Integration techniques
Lesson 10 of 11
Integration by substitution
Objective. Use a substitution to reverse the chain rule.
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.
- 1Let u = x² + 1, so du = 2x dx.
- 2Integral becomes ∫ u³ du = u⁴/4.
- 3Substitute back: (x² + 1)⁴/4 + c.
Answer: (x² + 1)⁴/4 + c
Practice
Each question comes with a picture and a listen button. Try it first, then reveal the answer.
Common mistake. Leaving the answer in terms of u for an indefinite integral.
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