Calculators / Calculus 2 / U-Substitution Calculator

U-Substitution Calculator

Published on: June 1, 2025

This U-Substitution Calculator helps you solve integrals using the u-substitution method and shows each step clearly. It works by choosing a suitable inner expression for u, rewriting the integral in simpler form, and then integrating with respect to the new variable before substituting back. This makes it useful for checking answers, understanding how the method works, and practising calculus step by step.

Step-by-step method

  1. Write the integral.
  2. Choose a substitution u = g(x).
  3. Compute du and rewrite in terms of u.
  4. Integrate with respect to u.
  5. Substitute back and add + C.

Formula:

Let u = g(x), so du = g′(x)dx
∫ f(g(x))·g′(x) dx = ∫ f(u) du
∫ xn dx =
xn+1
n+1
∫
1
x
dx = ln|x|
∫ sin(x) dx = −cos(x)
∫ cos(x) dx = sin(x)
∫ ex dx = ex

Example 1: 2x*cos(x^2), x

Step 1 - Write the integral.

In this problem: We want to rewrite it into a u-integral.

∫2xcos(x2) dx

Step 2 - Choose a substitution.

In this problem: Pick u as an inner function: u = x2.

u = x2

Step 3 - Differentiate u.

In this problem: Compute du = (du/dx) dx.

du = 2x dx

Step 4 - Rewrite the integral in u.

In this problem: Replace g(x) with u and g'(x)dx with du.

∫2xcos(x2) dx = ∫ cos(u) du

Step 5 - Integrate with respect to u.

In this problem: Compute an antiderivative in u.

∫ cos(u) du = sin(u) + C

Step 6 - Substitute back to x.

In this problem: Replace u with x2.

sin(u) → sin(x2)

Final answer: sin(x^2) + C

Example 2: x/(x^2+1), x

Step 1 - Write the integral.

In this problem: We want to rewrite it into a u-integral.

∫x
1
x2 + 1
 dx

Step 2 - Choose a substitution.

In this problem: Pick u as an inner function: u = x2.

u = x2

Step 3 - Differentiate u.

In this problem: Compute du = (du/dx) dx.

du = 2x dx

Step 4 - Rewrite the integral in u.

In this problem: Replace g(x) with u and g'(x)dx with du.

∫x
1
x2 + 1
 dx = ∫
1
2
1
u + 1
 du

Step 5 - Integrate with respect to u.

In this problem: Compute an antiderivative in u.

∫
1
2
1
u + 1
 du =
1
2
ln(2u + 2) + C

Step 6 - Substitute back to x.

In this problem: Replace u with x2.

1
2
ln(2u + 2) →
1
2
ln(2x2 + 2)

Final answer: log(2x^2 + 2)/2 + C