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Online Integral Calculator

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Wolfram|Alpha is a great tool for calculating antiderivatives and definite integrals, double and triple integrals, and improper integrals. The Wolfram|Alpha Integral Calculator also shows plots, alternate forms and other relevant information to enhance your mathematical intuition.

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Use Math Input above or enter your integral calculator queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask for an integral using plain English.

• integrate x/(x-1)
• integrate x sin(x^2)
• integrate x sqrt(1-sqrt(x))
• integrate x/(x+1)^3 from 0 to infinity
• integrate 1/(cos(x)+2) from 0 to 2pi
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What are integrals?

Integration is an important tool in calculus that can give an antiderivative or represent area under a curve..

The indefinite integral of , denoted , is defined to be the antiderivative of . In other words, the derivative of is . Since the derivative of a constant is 0, indefinite integrals are defined only up to an arbitrary constant. For example, , since the derivative of is . The definite integral of from to , denoted , is defined to be the signed area between and the axis, from to .

Both types of integrals are tied together by the fundamental theorem of calculus. This states that if is continuous on and is its continuous indefinite integral, then . This means . Sometimes an approximation to a definite integral is desired. A common way to do so is to place thin rectangles under the curve and add the signed areas together. Wolfram|Alpha can solve a broad range of integrals

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• Symbolab is the best integral calculator solving indefinite integrals, definite integrals, improper integrals, double integrals, triple integrals, multiple integrals, antiderivatives, and more.
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• Integration is a way to sum up parts to find the whole. It is used to find the area under a curve by slicing it to small rectangles and summing up thier areas.

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Integral Calculator

Enter the function you want to integrate into the editor.

Integration by parts formula : ? u d v = u v - ? v d u

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INTEGRATION

This tutorial begins with a discussion of antiderivatives, mathematical objects that are closely related to derivatives. After the integral is introduced via the area problem, the integral and the antiderivative are shown to be related by an amazing theorem called the fundamental theorem of calculus. After establishing some techniques for evaluating integrals, we exhibit the important interpretation of the integral as a limit of a certain sum and demonstrate a variety of applications of the integral to problems in business and economics, geometry, and science.

Antiderivatives

Suppose that during the initial stages of production the marginal cost to produce a commodity is C '(x) = 2x dollars per unit. This time, suppose the manufacturer also knows that the fixed cost of production, C(0), is $500. Find the corresponding cost function C (x). We have already seen that any cost function for this marginal cost must be of the form C(x) = x 2 + a for some constant a. Since

C (0) = 500 = 0 2 + a = a,

we have a = 500. Thus, the cost function is given by C(x) = x 2 + 500

From this example, we see that the arbitrary constant c is the fixed cost of production. Knowing only the marginal cost cannot tell us what that fixed cost is; the fixed cost is additional information. Each of the cost functions corresponding to a marginal cost of C'(x) = 2x will have the form

To verify Rule (3), recall that

Example 5 Use Rule (2) to evaluate each antiderivative:

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Integral Calculator

Solve definite and indefinite integrals (antiderivatives) of a function with our integral calculator. Our goal is to not only solve integral calculations but provide clarity on the process behind each integration step and reinforcing student understanding. In our experience, getting the answer is good, but showing your work is better! The integral calculator will handle both definite and indefinite integrals. While definite integrals quantify the net area beneath a curve within a specified range, indefinite integrals, also known as antiderivatives, depict a family of functions without set boundaries. But this tool isn't restricted to single-variable functions. We built it to handle functions with multiple variables. Just configure your calculation to handle the type of variable, add the upper and lower bound, and when you’re completely done, calculate it! Our goal is to build an integral calculator that will give you the right solution, but also help you grow as a student. Please reach out if you find any mistakes or need further clarification.

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Evaluating Integrals

An Integration calculator allows you to learn the concepts of calculating integrals without spending too much time. You can evaluate the integral using an integral calculator with steps easily online.

Similarly, you can find a double integral calculator on this website. The double integral calculator shows you graphs, plots, steps, and visual representation, which helps you learn advanced concepts of double integration.

There are many other useful calculators you can use to get benefit. Similarly, you can determine the volume of a solid of revolution with a washer method calculator and determine the cross sections of a solid of revolution with a disc method calculator .

How to use the Integral calculator?

Using the integration by parts calculator is easy and fast.

Follow these steps:

Step 1: Enter the function

To evaluate the integrals, you must have a proper function. You need to enter your function in the function bar of the integration calculator. There is also a "load example" list. You can click that list to load an example equation for calculating integrals step by step.

Step 2: Select the Variable

For evaluating integrals, there are three variables you can use. These variables are x, y and z. The role of these three variables differs from each other, and all 3 have different impacts on the overall output. You can select the variables as x, y and z from the variable section.

Step 3: Give Upper bound value

The upper bound is the value that helps us sum integral at its maximum value. The upper bound is denoted as U, and its determination is crucial in the integration process. You can enter the upper bound of your limit in the upper bound section of the upper bound calculator.

Step 4: Give Lower bound value

The lower bound is the smallest value that we set to start the integration. To get the accurate integration results, the smallest value of an interval is denoted by L. To get the precise integration results. You need to enter the actual amount of your lower bound limit in the lower bound section of the upper and lower bounds calculator.

After completing all above steps Press "GO" button.

Immediately after clicking the button our integral calculus calculator will start working. The integration by parts calculator will show you the anti derivative, integral steps, parsing tree and plot of your result. All these functionalities and features makes this the best line integral calculator to evaluate the integral of complicated integration problems.

What are integrals

An integral in calculus refers to a fundamental concept that is essentially the reverse process of differentiation. There are two key types of integrals we teach our students: definite and indefinite.

1. Indefinite Integral (Antiderivative):

The indefinite integral of a function represents a family of functions whose derivatives are the given function. Mathematically, it is represented as:

\[ \int f(x) \, dx = F(x) + C \]

where \( F(x) \) is an antiderivative of \( f(x) \) and \( C \) is an arbitrary constant.

2. Definite Integral:

The definite integral of a function over a specific interval \([a, b]\) calculates the net area between the function and the x-axis within that interval. This area is accounted for in both directions, meaning areas above the x-axis are treated as positive, while those below are treated as negative. It's mathematically represented as:

\[ \int_a^b f(x) \, dx \]

In layman's terms:

• Integrals can be visualized as the area under a curve .
• They represent an accumulation or total of quantities. For instance, integrating a velocity function gives a distance (or displacement) function.
• In physics, integrals are used in diverse scenarios such as calculating work done by a force or the distribution of charges in electric fields.

The process of finding integrals is called integration .

Understanding Integration

To solve for a definite integral, you have to understand first that definite integrals have start and endpoints, also known as limits or intervals, represented as (a,b) and are placed on top and bottom of the integral.

We can generalize integrals based on functions and domains through which integration is done. Integration by parts calculator with steps helps you to evaluate the integrals digitally.

Also: You can find the Line Integral Calculator and Surface Integral Calculator for more Information.

The formula for integral (definite) goes like this:

$\int_b^a f(x)dx=-\int_b^a f(x)dx$

$\int_b^a f(x)dx$

∫ represents integral

dx represents the differential of the 'x' variable

fx represents the integrand

point a and b represent limits of integration

Let's solve it considering that we're being asked for integral from 1 to 3, of 3x dx

$\int_3^1 3(x)dx$

$-\int_3^1 3(x)dx$

Take out the constant:

$\int a.f(x)dx = a.\int f(x)dx$

Applying the power rule :

$\int x^a dx = x^a+1/a+1, a≠1$

$=-3[\frac{x^{1+1}}{1+1}]_1^3$

$-3[\frac{x^2}{2}]_1^3$

calculating the boundaries: 4

On the other hand, the Indefinite integral is distinguished from the definite integral because of the former’s lack of defined limits.

Indefinite Integral goes by the formula:

$\int f(x)dx$

The above integration solver can calculate indefinite integral and definite integral, but if you want to calculate indefinite integral only, find the best online indefinite integral calculator .

Related:  Learn about definite integral and indefinite integral

How to calculate improper Integral?

One of the reasons why a definite integral becomes an improper integral is when one or both of the limits reach infinity. An Integral calculus calculator can be used to calculate improper integrals .

This integral is then solved by turning it into a problem of limits where c happens to approach infinity or negative infinity.

Let's consider an example where one of the limits of integration is infinite and then solve it.

$\int_1^\infty \frac{1}{x^2} dx \;and\; \int_1^\infty \frac{1}{x} dx$

$\int_1^\infty \frac{1}{x^2} dx = \lim_{c\to \infty} \int_1^c \frac{1}{x^2} dx$

$\lim_{c\to \infty} [-\frac{1}{x}]_1^c$

$\lim_{c\to \infty} [-\frac {1}{c}] -(-\frac{1}{1})]$

Since the answer to the improper integral is finite, we consider it converged.

If you only want to evaluate definite integrals, use this best step by step definite integral calculator online.

Related: Use shell method calculator with steps to find the volume of a solid of revolution easily online.

How to calculate Continuous Integration?

The fundamental theorem of calculus establishes a clear association between integral and differential calculus. Our integral calculator with steps is capable enough to calculate continuous integration.

If f(x) is continuous for the interval a and b given the variable x and G(x) is a function in such sense that dG/dx = f(x) for all values of x in (a,b)

Let f be continuous on an interval ‘y’. Select a point p in y then the function f(x) is defined as:

Let F(x) be as follows

$\int_p^x f(t) dt$

Let c be in i and let x be infinitely close to c and the endpoints of i. Then by addition,

$\int_p^c f(t) dt = \int_p^x f(t) dt + \int_x^c f(t)dt$

$\int_p^c f(t) dt - \int_p^x f(t) dt + \int_x^c f(t)dt$

$f(c) - F(x)= \int_x^c f(t)dt$

For your ease and advance learning regarding multiple integrals, we offer one of the quickest triple integral calculator . This tool will surely assist you in calculating definite and indefinite triple integrals online by doing few clicks.

Related: Understanding Integration by Partial Fraction in 5 Minutes!

Learning about integral calculations

How to evaluate integrals.

There are two types of integrals, definite and indefinite integrals. You can solve them both by integration. The difference is that you need to put the limit values after integration in definite integrals, whereas in indefinite integrals, you don’t need to put limit values.

Integrals calculator helps to solve out every type of definite and indefinite problems easily.

What is the integral of ex?

The integral of ex is:

$ \int e^x dx \;=\; \frac{e^x}{1}+c $ $ \int e^x dx \;=\; e^x+c$

The integration of exponential functions is tricky, but we provide great tools to evaluate integral online.

How to Evaluate the Integral by Interpreting it in terms of Areas?

Solving integration online are basically finding the area under a specific curve. For example of a given equation of curve 1-x with upper and lower bound x=-4 and x=3, the area will be calculated as,

$ \int_{-4}^3 (1-x) dx \;=\; \Biggr| x - \frac{x^2}{2} \Biggr|_{-4}^3 $ $ \int_{-4}^3 (1-x) dx \;=\; \left( 3 - \frac{3^2}{2} \right) \;-\; \left( -4 - \frac{(-4)^2}{2} \right) $ $ \int_{-4}^3 (1-x) dx \;=\; \left( 3 - \frac{9}{2} \right) \;-\; \left( -4 -2 \right) \;=\; \frac{21}{2} $

So the area under the given curve is 21/2. We can verify this by evaluate the integral calculator for cross-checking your answer.

The calculator integral is the great resource for this type of calculations to save your time.

What is the integral of 1/x?

The integral of 1/x is,

$ \int \frac{1}{x} dx \;=\; ln(x) + c$

Integral Examples & Functions

Integration terms and concepts.

Function : A relation from a set of inputs to a set of possible outputs where each input is related to exactly one output.

Limit : The value that a function approaches as the input approaches a certain value.

Continuous Function : A function for which small changes in the input result in small changes in the output, meaning it has no gaps, jumps, or holes.

Antiderivative : A function whose derivative gives the original function. It's the reverse process of differentiation.

Indefinite Integral : The antiderivative of a function, representing a family of functions.

Definite Integral : Represents the signed area under the curve of a function between two limits.

Integrand : The function being integrated.

Limits of Integration : The values a and b in a definite integral, representing the interval over which the function is integrated.

Constant of Integration : When determining the indefinite integral, there's an arbitrary constant C that's added because any constant's derivative is zero.

Fundamental Theorem of Calculus : Connects differentiation and integration. It states that if a function is continuous over an interval [a, b] and F is an antiderivative of f on [a, b] , then:

Integration by Substitution : A method used to transform a complicated integrand into a simpler one.

Integration by Parts : A technique based on the product rule of differentiation to integrate products of functions.

Partial Fractions : Decomposing rational functions to simpler fractions, making them easier to integrate.

Improper Integral : Integrals where one or both of the limits of integration are infinite, or where the integrand is unbounded.

Riemann Sum : An approximation of the definite integral using rectangles.

Disk/Washer Method : Techniques to find the volume of solids of revolution.

Cylindrical Shells Method : Another method to compute the volume of solids of revolution.

Integration of Trigonometric Functions : Special techniques and identities for integrating functions like sin(x), cos(x), tan(x), etc.

Integration of Exponential and Logarithmic Functions : Methods and formulas to integrate functions like e^x, ln(x), etc.

Integration of Hyperbolic Functions : Techniques for integrating functions like sinh(x), cosh(x), etc.

Linearity of the Integral : The principle that the integral of a sum is the sum of the integrals.

Sequences and Series : For integrating functions represented as power series or when using techniques like integration by Taylor series.

Convergence : A property that determines whether a sequence or series approaches a finite value.

When to use different integral calculators?

One of the biggest challeges is understanding when in how to calculate integrals. Each calculator is tailored to a specific kind of problem, and in my experience, students need to become as good at understanding what calculator to use as they are with solving integrals. Here's how I help my students understand which calculators to use: 

• Use When : You want to evaluate general integrals, either definite or indefinite.
• Purpose : Calculate the area under a curve, find antiderivatives, or accumulate quantities. Basic, go-to calculator for most problems in the classroom
• Use When : You want to evaluate integrals over a two-dimensional region.
• Purpose : Often used to find areas, volumes, or other quantities in multivariable calculus. A bit more advanced for my students!
• Use When : You're integrating over a three-dimensional region.
• Purpose : Useful for determining volumes and other three-dimensional quantities in multivariable calculus. This is really advanced. Used in engineering more than the classroom.
• Use When : You want to find the net area under a curve between two specific points.
• Purpose : To calculate a specific value representing the area between the function and the x-axis over a given interval.
• Use When : You want to find the general antiderivative of a function without specific bounds.
• Purpose : Outputs a function (or family of functions) representing the integral, typically including a constant of integration.
• Use When : You're finding the volume of a solid of revolution about a non-central axis using cylindrical shells.
• Purpose : It's a technique for volume when the disc/washer method is not applicable. You'll have to use your best understanding to know when the Shell method is appropriate
• Use When : You're determining the volume of a solid of revolution about an axis using washers (annular discs).
• Purpose : Used when the solid has a hole in the middle, like a donut shape.
• Use When : You aim to find the volume of a solid of revolution about an axis using discs.
• Purpose : Useful for determining volumes of solids with no holes in them, like a cylinder or cone.
• Use When : You want to transform a given function from the time domain to the frequency (s) domain.
• Purpose : Laplace Transforms are instrumental in solving linear ordinary differential equations, analyzing circuits, and other applications in engineering and physics

Alan Walker

Last Updated: 6 months ago

I am Mathematician, Tech geek and a content writer. I love solving patterns of different math queries and write in a way that anyone can understand. Math and Technology has done its part and now its the time for us to get benefits from it.

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Integral Calculator - Solve Definite and Indefinite Integrals (antiderivatives)

Input recognizes various synonyms for functions like asin , arsin , arcsin , sin^-1

Multiplication sign and brackets are additionally placed - entry 2sinx is similar to 2*sin(x)

List of mathematical functions and constants :

• ln(x) — natural logarithm

• sin(x) — sine

• cos(x) — cosine

• tan(x) — tangent

• cot(x) — cotangent

• arcsin(x) — arcsine

• arccos(x) — arccosine

• arctan(x) — arctangent

• arccot(x) — arccotangent

• sinh(x) — hyperbolic sine

• cosh(x) — hyperbolic cosine

• tanh(x) — hyperbolic tangent

• coth(x) — hyperbolic cotangent

• sech(x) — hyperbolic secant

• csch(x) — hyperbolic cosecant

• arsinh(x) — inverse hyperbolic sine

• arcosh(x) — inverse hyperbolic cosine

• artanh(x) — inverse hyperbolic tangent

• arcoth(x) — inverse hyperbolic cotangent

• sec(x) — secant

• csc(x) — cosecant

• arcsec(x) — arcsecant

• arccsc(x) — arccosecant

• arsech(x) — inverse hyperbolic secant

• arcsch(x) — inverse hyperbolic cosecant

• |x| , abs(x) — modulus

• sqrt(x) , root(x) — square root

• exp(x) — e to the power x

• conj(z) — \(\overline{z}\)

• a+b — \(a+b\)

• a-b — \(a-b\)

• a*b — \(a\cdot b\)

• a/b — \(\dfrac{a}{b}\)

• a^b , pow(a,b) — \(a^b\)

• sqrt7(x) — \(\sqrt[7]{x}\)

• sqrt(n,x) — \(\sqrt[n]{x}\)

• lg(x) — \(\log_{10}\left(x\right)\)

• log3(x) — \(\log_3\left(x\right)\)

• log(a,x) — \(\log_a\left(x\right)\)

• ln^2(x) , ln(x)^2 — \(\ln^2\left(x\right)\)

• y''' , y'3 — \(y'''\)

• d^2y/dx^2 , d2y/dx2 — \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\)

• lambda — \(\lambda\)

• Calculators
• Calculators: Calculus II
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Integral Calculator

Find indefinite integrals (antiderivatives) step by step.

This online calculator will try to find the indefinite integral (antiderivative) of the given function, with steps shown. Different techniques are used: integration by substitution, integration by parts, integration by partial fractions, trigonometric substitutions, etc.

Related calculator: Definite and Improper Integral Calculator

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

The Integral Calculator is an indispensable online assistant for working with indefinite integrals. Our user-friendly digital platform lets you calculate integrals and also provides step-by-step solutions to improve your skills in taking integrals.

How to Use the Integral Calculator?

Begin by entering the function you want to integrate. You can input a wide range of functions, from simple to complex.

Calculation

Once you've entered the function and, if necessary, the variable, click the "Calculate" button.

The calculator will immediately display (if it can calculate) the value of the integral.

What Is an Integral?

Indefinite Integral (Antiderivative)

The indefinite integral, often called the antiderivative, doesn't have specific bounds. It is an operation of finding such a function $$$ F(x) $$$ whose derivative equals a given function $$$ f(x) $$$ . Mathematically, the indefinite integral can be written as follows:

• $$$ f(x) $$$ is the integrand, i.e. the function to be integrated.
• $$$ F(x) $$$ is the antiderivative of $$$ f(x) $$$ .
• $$$ C $$$ is the constant of integration, which accounts for all possible antiderivatives: if $$$ F^{\prime}(x)=f(x) $$$ , then $$$ \left(F(x)+C\right)^{\prime}=f(x) $$$ .

For example, let's find the antiderivative of $$$ f(x)=2x $$$ . This can be written as follows:

There are many techniques for taking integrals, but this integral requires the constant multiple rule and the power rule:

So the antiderivative of $$$ 2x $$$ is $$$ x^2+C $$$ , where $$$ C $$$ represents the constant of integration.

Definite Integral

The definite integral, denoted by $$$ \int_a^b f(x)dx $$$ , represents the accumulated quantity or the net area between the curve of the function $$$ f(x) $$$ and the x-axis over the interval $$$ [a, b] $$$ . According to the Fundamental Theorem of Calculus, there is a connection between the definite integral and the antiderivative:

• $$$ a $$$ and $$$ b $$$ are the lower and upper bounds of the integration.
• $$$ f(x) $$$ is the integrand, representing the function to be integrated.
• $$$ F(x) $$$ is the antiderivative (indefinite integral) of $$$ f(x) $$$ .

For example, let's find the area below the curve $$$ f(x)=x^2 $$$ and above the x-axis between $$$ x=1 $$$ and $$$ x=3 $$$ . We can use the definite integral:

To evaluate this, we first find the antiderivative of $$$ x^2 $$$ , which is $$$ \frac{x^{2+1}}{2+1}=\frac{x^3}{3} $$$ . Now, we apply the Fundamental Theorem of Calculus:

So the area under the curve $$$ f(x)=x^2 $$$ between $$$ x=1 $$$ and $$$ x=3 $$$ is $$$ 26 $$$ square units.

The integral is an important tool to solve various mathematical problems. The indefinite integral helps us understand the relationship between the function and its antiderivative, and the definite integral allows us to calculate areas, accumulate quantities, etc. over certain intervals.

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Step-by-Step Solutions

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What are the common integration techniques?

• integration by substitution
• integration by parts
• integration by partial fractions
• trigonometric substitutions

Can you take numbers out of an integral?

Yes, you can take constants or numbers out of an integral. This property is known as the constant multiple rule of integration. When you have an integral of the form $$$ \int cf(x)dx $$$ , where $$$ c $$$ is a constant or any expression that does not depend on $$$ x $$$ , you can factor $$$ c $$$ out of the integral: $$$ \int cf(x)dx=c\int f(x)dx $$$ . It allows you to simplify the integration process when dealing with constants.

Can an integral be infinite?

Yes, a definite integral can be infinite under certain conditions. For example, if you have an improper integral with unbounded integration limits, or if the integrated function has an infinite discontinuity within an interval of integration, the integral may diverge, i.e. it may not have a finite value. For example, $$$ \int_{-1}^1\frac{dx}{x} $$$ diverges.

Can an integral be zero?

Yes, an integral can be evaluated to zero. When the area under a curve, as calculated by the integral, is equal to zero, it means that the positive and negative areas cancel each other out within the given interval. For example, $$$ \int_{-2}^2x^3dx=0 $$$ .

What is a double integral?

The double integral is an extension of the concept of integration to two-dimensional space. It involves integrating a two-variable function over a specified region in the xy-plane. Double integrals are used to calculate areas, volumes, and other quantities in two-dimensional space. The notation for a double integral is $$$ \iint f(x,y)dA $$$ .

What is an integral value?

There are two types of integral values: the indefinite integral value and the definite integral value. The indefinite integral value represents the result of integrating a function f(x) with respect to the variable $$$ x $$$ . It is such function $$$ F(x) $$$ that $$$ F^{\prime}(x)=f(x) $$$ . The definite integral value is the value of the integral over a specified interval. It can be a numerical value or some expression. The definite integral value provides information about quantities like area, accumulated total, or displacement, depending on the context of the problem.