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Choosing the substitution
Integration · Lecture 4
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Integration — Lecture notes
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Remark · page 3
Integration by substitution
Check whether part of the integrand is the derivative of another part. If it is, a substitution may simplify the integral.
∫ (2x + 1) / √(x² + x + 1) dx
Hint: the numerator is the derivative of the radicand.
02
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Practice questions
Paper 1 · Integration
Which substitution is most efficient for
∫ (2x + 1) / √(x² + x + 1) dx
- Au = x² + x + 1
- Bu = 2x + 1
- CIntegration by parts
- DPartial fractions
- ETrig substitution
3 of 8 completedCheck answer
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Worked solution
Shown step by step
Step 1 · Recognise structure
Numerator is d/dx of the radicand.
Step 2 · Substitute
Let u = x² + x + 1, so du = (2x + 1) dx.
Step 3 · Integrate
∫ u^(-1/2) du = 2√u + C
Final answer
2√(x² + x + 1) + C
04
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This sample is from the Paper 1 integration module.
Lesson video
A lesson from the Paper 1 integration module.
Integration by substitution
Paper 1 · Sample lecture
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