U Substitution Calculator

Use our u substitution calculator to simplify and solve integrals easily. It provides step-by-step solutions for both definite and indefinite integrals.

Definite Integral
Indefinite Integral

with respect to

Upper Bound

Lower Bound

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Introduction to U Substitution Calculator:

U Substitution Calculator with steps is an online tool that helps you evaluate both definite or indefinite integral problems in a fraction of a second. Our tool determines the complex structure of integral function into a simplified form to make the integration process easy.

U Substitution Calculator with steps

The integration by substitution calculator is a helpful tool for everyone who wants to solve the complicated form of an integral function without doing complex calculations because our tool simplifies the given integral function with the help of the u-substitution method.

What is U Substitution?

U substitution is a process that is used to convert the complex structure of a given integral function into simplify form to make the integration process smooth and easy. After evaluating the integral replace the u value with its original function value.

You can use this method to get a solution to both types of integral, definite or indefinite integral. The notation that the u sub calculator uses to express u-substitution is given as,

$$ \int f[g(x)] g’(x) dx \;=\; \int f(u) du $$

Where g(x) is the integral function and g`(x) is the derivative of the given function f(u) is the supposed value and du is replaced by dx when you differentiate it with respect to u.

The integration solution we get is,

$$ =\; F(u) + C $$

$$ =\; F(g(x)) + C $$

How to Do U Substitution?

U substitution method can make the integration process easy and you get the solution of the integral function without taking a difficult method for integration. Let us see how the u substitution calculator with steps calculates the integral problems with an example. The stepwise calculation for u substitution is:

Example: Find the integral

$$ \int z \sqrt{z^2 - 5} dz $$

Solution:

Step1:

As you can see the given function has a complex form so the integral substitution calculator uses the u-substitution method.

Step2:

For that suppose the given values in the function are equal to u and du form,

$$ u \;=\; z^2 - 5 $$

$$ du \;=\; 2zdz $$

$$ \frac{1}{2} du \;=\; \frac{1}{2}(2z) dz \;=\; z\; dz $$

Step3:

Put these values in the given function,

$$ \int z(z^2 - 5)^{\frac{1}{2}} dz \;=\; \frac{1}{2} \int u^{\frac{1}{2}} du $$

Step4:

Evaluate the integral of the above function as you can see the above integral function becomes the simplest function of integration after using the u-substitution method. Apply the power rule of integration such as,

$$ \frac{1}{2} \int u^{\frac{1}{2}} du \;=\; \left( \frac{1}{2} \right) \frac{u^{\frac{3}{2}}}{3} + C $$

$$ =\; \left( \frac{1}{2} \right) \left(\frac{2}{3} \right) u^{\frac{3}{2}} + C $$

$$ =\; \frac{1}{3} u^{\frac{3}{2}} + C $$

Step 5:

Now replace u with the given original function value for the solution.

$$ =\; \frac{1}{3} (z^2 - 5)^{\frac{3}{2}} + C $$

Step 6:

If the given function is a definite integral function then the u-substitution method is the same but after integration, you can apply the limits to get a solution.

The usub calculator do this all procedure and gives an updated answer which solves your all queries about the question.

How to Use the U Substitution Calculator?

The integration by substitution calculator has a user-friendly design that allows you to give different types of integration functions to get solutions immediately.

You just need to put your input function in this u sub calculator by following some simple steps that help you to get results without any inconvenience. These steps are:

Enter the integral function that you want to evaluate for u-substitution in the input field
Add the variable of integration
If the function is a definite integral function then add the upper and lower limit values in the input field.
Review your given input value before clicking the calculate button to get the solution of the integral function.
Click the “Calculate” button for the solution of integral problems in the integral substitution calculator.
If you want to check the working process behind the substitution integral calculator then use the load example for the calculation of integral.
Click the “Recalculate” button for the evaluation of more examples of the u-substitution integral problem.

Results from Integration by Substitution Calculator:

U substitution calculator with steps provides you with a u-substitution problem solution as per your input value when you click on the calculate button. It may include as:

In the Result Box:

When you click on the result button you get the solution of the u-substitution integral problem.

Steps Box:
Click on the steps option so that you get the solution to your u substitution integral questions in a stepwise method.

Benefits of Using U Sub Calculator:

The usub calculator has multiple benefits that you can avail whenever you use it to get a solution of u-substitution integration problems and get faster solutions. These benefits are:

Our u-substitution calculator only takes the input value of the integral function and gives a solution without imposing a condition of a sign-up option.
It is an unlimited tool which means you can use it as many times as you can for calculation.
The integration by substitution calculator is a trustworthy tool as it always provides you with accurate solutions for the substitution of integral problem
It is a speedy tool that evaluates the integral problems with the help of u-substitution and gets solutions in a couple of seconds
The integral substitution calculator is a learning tool that helps children about the concept of integral function very easily in an online platform without going to any teacher.
The substitution integral calculator is a free tool that allows you to use it for the calculation of u-substitution integration problems without spending a single penny.
It is an easy-to-use tool, anyone even a beginner can easily use it for the solution of integral problems of the u-substitution method.
U substitution calculator with steps can operate on a desktop, mobile, or laptop through the internet to solve integral-function problems.

Related References

Frequently Ask Questions

When to use u-substitution

U-substitution is the best technique in integral calculus that is used to simplify the process of finding the antiderivative of a function. Here are some steps to understand when to use u-substitution:

When to Use u-Substitution

When the integrand is a composite function f(g(x)), where you can identify an inner function g(x) and its derivative g′(x) within the integrand.
When the integrand is a product of a function and its derivative, or a function
When the composite integral function and its derivative are present in the form f(g(x)), g ′(x), then you can use this substitution.
When integrating exponential functions where the exponent is a linear function of x, or logarithmic functions involving 1/x
When the integrand involves trigonometric functions where the argument is a linear function of x.

How to find u substitution vs integration by part

To determine whether to use u-substitution or integration by parts depends on the structure of the integrand. Here’s how to decide which technique to use and how to apply each method:

U substitution and Integration by parts

U-substitution is used when the integrand includes a function and its derivative. This technique is useful when the integral is of the form ∫f(g(x))g′(x)dx

Steps for u-Substitution

Choose u=g(x), where g(x) is a part of the integrand. And compute du = g′(x)dx.
Substitute u and du values in the integral.
Integrate it with respect to u.
Replace u with g(x) to express the answer in terms of x.

To use integration by parts when the integrand is a product of two functions, typically one that becomes simpler when differentiated and another that is easily integrated. This technique is based on the formula which is

$$ \int u\; dv \;=\; uv − \int v\; du $$
Steps for Integration by Parts
Suppose u = f(x) and dv = g(x) dx and du = f′(x) dx
Integrate dv as v = ∫ g(x)dx.
Put your supposed value in integration by parts formula to get solution.

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

If you choose u-substitution for integrals then a function has its derivative or if you use integration by parts for integrals that involve a product of functions where one can be easily differentiated and the other easily integrated. The structure of the integrand will determine the appropriate technique and apply it effectively to solve the integral.

What is the purpose of U-substitution

The purpose of u-substitution is to simplify the process of evaluating integrals, especially when it deals with composite functions or integrals that contain a function and its derivative. It is a process that transforms a more complex integral into a simpler one for appropriate substitution to get a given function solution.

The main objective of u-substitution is to change the variable of integration from x to a new variable u such that the integrand becomes easier for integration. This is achieved when you choose u=g(x) where g(x)is a part of the integrand that, when substituted, simplifies the integral.

What is the difference between T substitution and U-substitution?

Trigonometric substitution and U-substitution are both techniques that are used to simplify and evaluate the given integrals. Both methods are applied in different contexts and for different types of integrals. Here’s a detailed comparison to understand the differences and when to use each method:

u-Substitution

When the integrand includes a function and its derivative, or when it is a composition of functions. Then U substitution is used in which u = g(x) to simplify the integral, where g(x) is part of the integrand.

Calculation steps of U substitution

Identify the substitution u = g(x)
Compute the differential du = g′(x)dx.
Rewrite the integral in terms of u and du
Integrate with respect to u.
Replace U by substituting the original variable in the solution.

Trigonometric Substitution

It is used to simplify integrals involving square roots of quadratic expressions, specifically those of the forms √a²−x². It uses trigonometric identities to substitute for x in such a way that the integrand changes into a trigonometric integral for evaluation

Calculation steps of Trigonometric substitution

Identify the given integral function expression under the square root.
Choose the type of trigonometric substitution according to a given function

For √a² − x², use x = a sin⁡(θ) x , x = a sin(θ).
For √a² + x², use x = a tan⁡(θ) x , x = a tan(θ).
For √x² + a^{2, use x = a sec⁡(θ)x , x = a sec⁡(θ)}

Substitute and simplify the integral using trigonometric identities and integrate with respect to the trigonometric variable.
Use the inverse trigonometric function to substitute back to the original variable to get the solution.

Evaluate the given integral.∫ 3x^2 cos(x^3) dx

Apply the U substitution,

So we let u = x^3 and then du = 3x^2

$$ \int 3x^2 cos(x^3) dx $$

$$ \int cos(u)\; du $$

$$ =\; sin(u) + C $$

Replace u with the original value,

$$ =\; sin(x^3) + C $$

Amanda Jepson

Last Updated February 21, 2024