u-Substitution Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of u-Substitution.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An integration technique where you substitute u = g(x) and du = g'(x)\,dx to transform a complicated integral into a simpler one. It is the reverse of the chain rule for differentiation.

When you see a composite function inside an integral along with its inner derivative lurking nearby, substitution collapses the composition into a single variable. It's like un-nesting a function: replace the inner part with u, and the integral becomes simpler.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: u-substitution reverses the chain rule. Since \frac{d}{dx}[F(g(x))] = F'(g(x)) \cdot g'(x), integrating F'(g(x)) \cdot g'(x) recovers F(g(x)).

Common stuck point: The trickiest part is choosing the right u. Look for a function whose derivative also appears in the integrand (perhaps off by a constant factor). The 'inner function' of a composition is usually a good first guess.

Sense of Study hint: Circle the inner function in the integrand as your u, then check whether du (or a constant multiple of it) appears in the rest.

Worked Examples

Example 1

easy
Find \displaystyle\int 3x^2(x^3+1)^4\,dx.

Solution

  1. 1
    Let u = x^3+1, so du = 3x^2\,dx.
  2. 2
    Integral becomes \int u^4\,du = \frac{u^5}{5} + C.
  3. 3
    Substitute back: \frac{(x^3+1)^5}{5} + C.

Answer

\frac{(x^3+1)^5}{5} + C
3x^2\,dx is exactly du, so the substitution is perfect. After substitution the integral reduces to a simple power rule.

Example 2

medium
Evaluate \displaystyle\int_0^1 xe^{x^2}\,dx.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find \displaystyle\int \cos(5x)\,dx.

Example 2

hard
Find \displaystyle\int \frac{\ln x}{x}\,dx.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

integralchain rule