Integral Calculator

The integral calculator, your go-to tool for solving integrals online. Whether it’s an indefinite or definite integrals, use it to get quick results.

Definite Integral
Indefinite Integral

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Introduction to Integral Calculator With Steps:

Integral calculator is an online tool that helps you evaluate the solution of integral for trigonometry, exponential, algebraic, or polynomial expression, and so on. It can also be used to integrate definite or indefinite functions.

Integral Calculator with Steps

When you calculate the integrals manually, you get stuck in getting the result as it has complex structure. To get rid from the calculations, you can try our integration calculator that easily gives you solutions to your integral questions, allowing you to integrate online and streamline your problem-solving experience.

What is an Integral?

Integral is a fundamental concept of calculus. It is a process of finding the infinitesimal data from the given function under a curve or a graph. This process is known as the inverse derivative process. It can be represented as “∫” in calculus.

Integral calculation is very important for performing complex tasks in daily life because it is used in physics, engineering, science, and mathematical fields. It has two main categories: definite integral and indefinite integral.

Indefinite Integral:

Indefinite integral is sometimes called a derivative function because it does not involve a limit. To calculate indefinite integral, following formula is used.

$$ \int f(x)\; dx \;=\; F(x) + C $$

Definite Integral:

Definite integral is the area that is present in a closed interval. Here a and b denote the lower and the upper limit respectively.

$$ \int_a^b f(x)\; dx $$

How to Calculate Integrals?

To calculate the integral you should know the basic integral rules that are used to solve various types of integral problems.

Let's see how to calculate the definite and indefinite integral with the help of an example so that you get clarity about the integral concept.

Definite Integral Calculation:

To calculate the definite integral, you need to follow these steps to be clear while solving the definite integral.

Identify the function f(x) and the limits of integration a and b.
Find the indefinite integral with respect to the given function F(x) to f(x).
Evaluate the indefinite integral at the Limits, after solving the upper limit b and the lower limit a onto the antiderivative function, $$ \int ab^f(x)\; dx \;=\; F(b) − F(a) $$

Let's see an example of a definite integral with steps.

Example:

Calculate the definite integral:

$$ \int_0^2 (3x^2 + 2x + 1)\; dx $$

Solution:

Take the integration of the given integral question,

$$ \int (3x^2 + 2x + 1)\; dx \;=\; x^3 + x^2 + x + C $$

Apply limits on the above result of an integral question,

$$ \int_0^2 (3x^2 + 2x + 1)\; dx \;=\; [x^3 + x^2 + x]_0^2 $$

$$ =\; (2^3 + 2^2 + 2) - (0^3 + 0^2 + 0) $$

$$ =\; (8 + 4 + 2) - (0 + 0 + 0) $$

$$ =\; 14 $$

Therefore the solution of the given definite problem is 14.

Indefinite Integrals Calculation:

To calculate an indefinite integral, follow these steps:

Identify the function f(x) to be integrated from the given integral problems. After knowing the type of integral, use standard rules to integrate common functions. These rules are:

Power Rule:

$$ \int x^n\; dx \;=\; \frac{x^{n+1}}{n+1} + C (for\; n≠−1) $$

Exponential Rule:

$$ \int e^x\; dx \;=\; e^x + C $$

Trigonometric Rules:

$$ \int sin(x)\; dx \;=\; -cos(x) + C $$

$$ \int cos(x)\; dx \;=\; sin(x) + C $$

$$ \int sec^2(x)\; dx \;=\; tan(x) + C $$

Substitution Method:

For more complex functions, you can use the substitution method to solve the integral problem.

Let u = g(x)(the given function) and change the integral in terms of u to make the integration process easy. Integrate with respect to u and again change u with x in the result of the given integral function.

Integrate by Parts

When you have an integration that has the products of functions, then you can use integration by parts method. The formula to solve the integration by parts related problems is,

$$ \int u dv \;=\; uv - \int v\;du $$

Choose u and dv such that the integral of vdu

Let's see an example of an indefinite integral with a solution so find the example of an indefinite integral function which is:

$$ \int (4x^3 - 2x + 5)\; dx $$

Solution:

Takes the integration of a given function,

$$ \int 4x^3 dx \;=\; 4 \int x^3 dx $$

The integral of xⁿ is,

$$ \frac{x^{n+1}}{n+1} \; \; \; n ≠ -1 $$

$$ \int x^3 dx \;=\; \frac{x^4}{4} $$

How to Use the Integration Calculator?

Integration solver has a simple design that helps you solve the given integral function question immediately. You just need to put your problem in it and follow some important instructions to get integration results without any trouble.

These steps ensure that integration online is easy and makes problem-solving straightforward. These instructions are:

Choose the type of integral from the input box of integral calculator.
Enter the function of the integral that you want to evaluate the integration solution in the input fields.
Recheck your given input value to get the exact solution of the integral question.
The Calculate button provides you with solutions to given integral problems.
If you want to check the accuracy of our integrals calculator, you can use the load example and get its solution.
Click the “Recalculate” button for the evaluation of more examples of the integrals with the solution.

Outcome of Integral Solver With Steps:

Integral online provides you with a solution as per your input problem when you click on the calculate button. It may include as:

In the Result Box,

Click on the result button for the solution of the integration question.

Steps Box
When you click on the steps option, you get the solution of integral questions in a step-by-step process.

Benefits of Integration Solver:

Integrals calculator has several benefits that you get whenever you use it to get the solution of integral questions. These benefits are:

Our integral calculator with steps is a trustworthy tool as it always provides you with accurate solutions of given integral problems.
It is an efficient tool that provides solutions in a step by step process of the given integral problems.
Integration calculator is a learning tool that helps you to easily grab the concept of integration very easily through an online platform.
It is a handy tool that solves integral problems with the help of different rules as per the given problems and you do not put any type of external effort.
Integral solver with steps is a free tool that allows you to use it for the calculation of integral problems without spending.
It is an easy-to-use tool, anyone, even a beginner can easily use it for the solution of integral problems.
Integral online can save you tons of time that you use in finding the solution to integral questions as it provides you results in a run of time.

Related References

Frequently Ask Questions

Does the Integral Converge or Diverge

To determine whether an integral converges or diverges, evaluate the integral using limits and check whether the behavior of the integrand is at infinity or near singularities. Then use the comparison tests, such as:

To compare f(x) from a known function g(x) such that 0 ≤ f(x) ≤ g(x) for all x in the interval.

If ∫g(x)dx converges and f(x) ≤ g(x), then ∫f(x)dx also converges. If ∫g(x) dx diverges and f(x) ≥ g(x), then ∫f(x)dx is also diverges.

What is the Integral of ln x

To find the integral of ln⁡(x), you can use integration by parts based on the formula:

$$ \int u\; dv \;=\; uv - \int v\; du $$

To get the solution of lnx, lets suppose

$$ u \;=\; ln (x) $$

$$ dv \;=\; dx $$

$$ du \;=\; \frac{d}{dx} (ln(x))\; dx \;=\; \frac{1}{x} dx $$

$$ v \;=\; \int\; dx \;=\; x $$

Put this substitution in integration by parts formula,

$$ \int\; ln(x)\; dx \;=\; x\; ln\; (x) - \int\; x . \frac{1}{x} dx $$

Simplify the above expression,

$$ \int\; x . \frac{1}{x} dx \;=\; \int 1\; dx \;=\; x $$

$$ \int\; ln\; (x)\; dx \;=\; x\; ln\; (x) - x\; + C $$

Therefore the solution for given integral problem is,

$$ \int\; ln\; (x)\; dx \;=\; x\; ln\; (x) - x\; + C $$

What is an Improper Integral?

An improper integral is an integral where the interval of the integral function is infinite or the integrand has an infinite discontinuity within the given interval. These integrals are called "improper" integrals because they do not meet the standard criteria that are necessarily involved in Riemann integrals, but they still can be evaluated by the limits.

Improper integrals involve limits to handle infinite limits with infinite discontinuities, when you take the limits, you can determine either the integral converges (finite value) or diverges (does not have a finite value).

What is the Integration of 1/x

The integration of 1/x is a basic rule of integral in calculus. Let's see how you can solve it.

Take the integral of 1/x with respect to x is given by:

$$ \int \frac{1}{x} dx \;=\; ln\; ⁡∣x∣ + C $$

where C is the constant of integration.

What is the Integral of Tanx

To find the integral of tan⁡(x), you can use the fact that tan⁡(x) can be expressed in terms of sine and cosine such as:

$$ tan\; (x) \;=\; \frac{sin (x)}{cos (x)} $$

So it can be written as,

$$ \int tan\; (x) dx \;=\; \int \frac{sin(x)}{cos(x)} dx $$

Lets solve the given function with u substitution,

$$ u \;=\; cos\; (x) . du \;=\; -sin\; (x) dx - du \;=\; sin (x)\; dx $$

Put these values in the above integral function,

$$ \int \frac{sin(x)}{cos(x)} dx \;=\; \int \frac{-du}{u} \;=\; -\int \frac{1}{u} du $$

Take the integral of above function with respect to u,

$$ -\int \frac{1}{u} du \;=\; -ln |u| + C $$

Again replace the u value with the given function and simplify the solution.

$$ u \;=\; cos(x): $$

$$ \int tan(x)\; dx \;=\; -ln |cos(x)| + C $$.

Hence the solution of tanx is,

$$ \int\; tan(x)\; dx \;=\; ln |sec(x)| + C $$

Amanda Jepson

Last Updated February 21, 2024