u-Substitution Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

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

Solution

  1. 1
    Let u=5xu = 5x, du=5 dxdu = 5\,dx, so dx=du5dx = \frac{du}{5}.
  2. 2
    15∫cos⁡u du=15sin⁡u+C=sin⁡(5x)5+C\frac{1}{5}\int \cos u\,du = \frac{1}{5}\sin u + C = \frac{\sin(5x)}{5} + C.

Answer

sin⁡(5x)5+C\frac{\sin(5x)}{5} + C
Linear inner functions are the simplest substitutions. The constant factor from dudu divides the result.

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 1 easy

Find [formula].

Example 2 medium

Evaluate [formula].

Example 4 hard

Find [formula].

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