Integration by substitution
Objective. Transform variables, integrands and limits consistently.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Integration by substitution transforms a difficult integral into a simpler one by changing the variable. Choose u as an inner function, find du/dx, rewrite dx in terms of du, and substitute throughout, including changing the limits for a definite integral. For ∫ 2x(x²+1)³ dx, let u = x²+1, so du = 2x dx, giving ∫ u³ du = u⁴/4 + c = (x²+1)⁴/4 + c. Always substitute every x-term, including the limits, before integrating.
Worked example
Use integration by substitution to find ∫ x√(x²+4) dx.
- 1Let u = x²+4, so du = 2x dx, giving x dx = du/2
- 2Rewrite the integral as ∫ √u × (du/2)
- 3Integrate: (1/2) × (2/3)u^(3/2) = (1/3)u^(3/2)
- 4Substitute back u = x²+4 to get (1/3)(x²+4)^(3/2) + c
Answer: (1/3)(x²+4)^(3/2) + c
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Practice
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Start practisingCommon mistake. A frequent error in definite integrals is integrating in terms of u but then substituting the original x-limits back in. Always convert the limits to u-values immediately after choosing the substitution, or convert back to x before evaluating.
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