u-Substitution Math Example 1

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Example 1

easy
Find ∫3x2(x3+1)4 dx\displaystyle\int 3x^2(x^3+1)^4\,dx.

Solution

  1. 1
    Let u=x3+1u = x^3+1, so du=3x2 dxdu = 3x^2\,dx.
  2. 2
    Integral becomes ∫u4 du=u55+C\int u^4\,du = \frac{u^5}{5} + C.
  3. 3
    Substitute back: (x3+1)55+C\frac{(x^3+1)^5}{5} + C.

Answer

(x3+1)55+C\frac{(x^3+1)^5}{5} + C
3x2 dx3x^2\,dx is exactly dudu, so the substitution is perfect. After substitution the integral reduces to a simple power rule.

About u-Substitution

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

Learn more about u-Substitution →

More u-Substitution Examples

Example 2 medium

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Example 3 easy

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Example 4 hard

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