Find The Integral Using U Substitution Calculator

This guide explains how to find integrals using u-substitution, a powerful technique in calculus. We provide a calculator to solve integrals quickly and a detailed explanation of the method.

What is U-Substitution?

U-substitution is a method for finding antiderivatives by reversing the chain rule. It's particularly useful for integrals involving composite functions. The basic idea is to substitute a part of the integrand with a new variable, solve the integral, and then substitute back.

∫f(g(x))·g'(x) dx = ∫f(u) du where u = g(x)

This technique allows us to simplify complex integrals into more manageable forms. The key is to identify the inner function and its derivative in the integrand.

How to Use the Calculator

Our calculator makes it easy to find integrals using u-substitution. Simply enter your integrand in the input field, select the substitution variable, and click "Calculate". The calculator will show you the step-by-step solution and the final result.

Example

To find ∫x²cos(x³ + 5) dx, you would:

Identify that u = x³ + 5
Find du/dx = 3x² (so du = 3x² dx)
Adjust the integral: (1/3)∫cos(u) du
Integrate to get (1/3)sin(u) + C
Substitute back: (1/3)sin(x³ + 5) + C

Step-by-Step Method

Identify the substitution: Choose u to be the inner function of the integrand.
Find du: Differentiate u with respect to x to find du.
Rewrite the integral: Express the original integral in terms of u and du.
Integrate: Solve the integral with respect to u.
Substitute back: Replace u with the original expression in terms of x.

Tip: When choosing u, look for a function whose derivative is also present in the integrand. This makes the substitution straightforward.

Common Integrals Solved with U-Substitution

Here are some common integrals that can be solved using u-substitution:

Integral	Substitution	Result
∫x cos(x² + 3) dx	u = x² + 3	(1/2)sin(x² + 3) + C
∫eˣ sin(eˣ + 2) dx	u = eˣ + 2	-cos(eˣ + 2) + C
∫sec²(3x) tan(3x) dx	u = tan(3x)	(1/3)sec²(3x) + C

FAQ

When should I use u-substitution?

Use u-substitution when the integrand is a composite function and its derivative appears elsewhere in the integrand. This method is particularly effective for integrals involving trigonometric, exponential, and logarithmic functions.

What if I can't find a suitable substitution?

If you can't identify a suitable substitution, try other integration techniques like integration by parts, trigonometric identities, or partial fractions. Sometimes, a combination of methods may be needed.

How do I know if I've done the substitution correctly?

Check that your substitution u = g(x) and its derivative du = g'(x) dx correctly transform the original integral into a simpler form. The integral should be expressed entirely in terms of u after substitution.