U-Substitution Calculator

Published on: July 18, 2026
Final Answer: Free • Full Steps: Plus

This U-Substitution Calculator integrates a function by renaming an inner expression, and shows the working line by line. A part of the integrand is called u, its derivative is matched against the rest, and the integral is rewritten entirely in u so that a standard rule applies. Substituting back at the end returns the answer in x. Following the same pattern every time keeps the change of variable reversible.

Step-by-step method

  1. Set up the problem.
  2. Choose the inner function u and compute du.
  3. Rewrite the whole integral in terms of u.
  4. Integrate with respect to u.
  5. Substitute back to x and add C.

Formula:

\(\int f\left(g\left(x\right)\right) g'\left(x\right)\, dx = \int f\left(u\right) \, du, \quad u = g\left(x\right)\)
See Example 1 Hide Example 1

Problem

\(\int 2 x \left(x^{2} + 1\right)^{5}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Choose the inner function u and compute du.
  3. Rewrite the whole integral in terms of u.
  4. Integrate with respect to u.
  5. Substitute back to x and add C.

Step 1

Step 1 - Set up the problem.

In this problem: We are integrating \(2 x \left(x^{2} + 1\right)^{5}\). It contains an inner function whose derivative also appears as a factor.

\(\int 2 x \left(x^{2} + 1\right)^{5}\, dx\)

Step 2

Step 2 - Write the u-substitution formula.

In this problem: When the integrand is a composite times the derivative of its inside, substitute u for the inside.

\(\int f\left(g\left(x\right)\right) g'\left(x\right)\, dx = \int f\left(u\right) \, du, \quad u = g\left(x\right)\)

Step 3

Step 3 - Choose the inner function u and compute du.

In this problem: Let \(u = x^{2} + 1\). Then \(du = 2 x \, dx\).

\(\begin{gathered} u = x^{2} + 1 \\ du = 2 x\, dx \end{gathered}\)

Step 4

Step 4 - Rewrite the whole integral in terms of u.

In this problem: The remaining factor becomes \(u^{5}\) in terms of u.

\(\int 2 x \left(x^{2} + 1\right)^{5}\, dx = \int u^{5} \, du\)

Step 5

Step 5 - Integrate with respect to u.

In this problem: Integrating gives \(\frac{u^{6}}{6}\).

\(\int u^{5} \, du = \frac{u^{6}}{6} + C\)

Step 6

Step 6 - Substitute back to x and add C.

In this problem: Replace u with \(x^{2} + 1\).

\(\int 2 x \left(x^{2} + 1\right)^{5}\, dx = \frac{\left(x^{2} + 1\right)^{6}}{6} + C\)

Final Answer

\(\int 2 x \left(x^{2} + 1\right)^{5}\, dx = \frac{\left(x^{2} + 1\right)^{6}}{6} + C\)
See Example 2 Hide Example 2

Problem

\(\int 2 x \cos{\left(x^{2} \right)}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Choose the inner function u and compute du.
  3. Rewrite the whole integral in terms of u.
  4. Integrate with respect to u.
  5. Substitute back to x and add C.

Step 1

Step 1 - Set up the problem.

In this problem: We are integrating \(2 x \cos{\left(x^{2} \right)}\). It contains an inner function whose derivative also appears as a factor.

\(\int 2 x \cos{\left(x^{2} \right)}\, dx\)

Step 2

Step 2 - Write the u-substitution formula.

In this problem: When the integrand is a composite times the derivative of its inside, substitute u for the inside.

\(\int f\left(g\left(x\right)\right) g'\left(x\right)\, dx = \int f\left(u\right) \, du, \quad u = g\left(x\right)\)

Step 3

Step 3 - Choose the inner function u and compute du.

In this problem: Let \(u = x^{2}\). Then \(du = 2 x \, dx\).

\(\begin{gathered} u = x^{2} \\ du = 2 x\, dx \end{gathered}\)

Step 4

Step 4 - Rewrite the whole integral in terms of u.

In this problem: The remaining factor becomes \(\cos{\left(u \right)}\) in terms of u.

\(\int 2 x \cos{\left(x^{2} \right)}\, dx = \int \cos{\left(u \right)} \, du\)

Step 5

Step 5 - Integrate with respect to u.

In this problem: Integrating gives \(\sin{\left(u \right)}\).

\(\int \cos{\left(u \right)} \, du = \sin{\left(u \right)} + C\)

Step 6

Step 6 - Substitute back to x and add C.

In this problem: Replace u with \(x^{2}\).

\(\int 2 x \cos{\left(x^{2} \right)}\, dx = \sin{\left(x^{2} \right)} + C\)

Final Answer

\(\int 2 x \cos{\left(x^{2} \right)}\, dx = \sin{\left(x^{2} \right)} + C\)

Example Problems

Click one to load it into the box and solve it.