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Integration by u-substitution worksheets Grade 12

Integrate by choosing u as the inside function and rewriting in u.

The front of the free integration by u-substitution worksheet
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Free · Worked example · 15 problems · Answer key · US Letter

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How to do it

Let u be the inside. Write du, rewrite the integral in u, integrate, then put u back.

∫sin⁡(3x) dx=−cos⁡(3x)3+C\int \sin (3x)\, dx = -\dfrac{\cos (3x)}{3} + C
u=3x,du =3 dxu = 3x,\quad \text{du }= 3\ \text{dx}divide by 3
−cos⁡(3x)3+C-\dfrac{\cos (3x)}{3} + C

Example problems

  1. ∫cos⁡(2x) dx=\int \cos (2x)\, dx =
  2. ∫cos⁡(5x) dx=\int \cos (5x)\, dx =
  3. ∫cos⁡(3x) dx=\int \cos (3x)\, dx =
  4. ∫cos⁡(6x) dx=\int \cos (6x)\, dx =

Integrals

  1. Antiderivatives with the power rule
  2. Integrals of 1/x, eˣ, sin and cos
  3. Integration by u-substitution
  4. Evaluating definite integrals
  5. Area under a curve
  6. Area between two curves
  7. Average value of a function