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MathidiaMathematics Academy
Integration techniques
Lesson 10 of 11
Standard practice

Integration by substitution

Objective. Use a substitution to reverse the chain rule.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Choose u to be the awkward inner function, replace dx using du = u′ dx, and change the limits too if the integral is definite.

Worked example

Find ∫ 2x(x² + 1)³ dx.

Find ∫ 2x(x² + 1)³ dx
Illustrated card showing Find ∫ 2x(x² + 1)³ dx
  1. 1Let u = x² + 1, so du = 2x dx.
  2. 2Integral becomes ∫ u³ du = u⁴/4.
  3. 3Substitute back: (x² + 1)⁴/4 + c.

Answer: (x² + 1)⁴/4 + c

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Common mistake. Leaving the answer in terms of u for an indefinite integral.

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