Integration Techniques — Topic Review

← Back to Integration Techniques

This review covers all four lessons in the Integration Techniques topic: Integration by Substitution, Integration by Parts, Partial Fractions Integration, and Integration with Trig Identities. Questions span all difficulty levels and mirror the style of QCAA Specialist Maths examination items.

Review Questions

1. Fluency

   Q1 — Substitution: power of a function

   Find ∫4x(x² + 5)³ dx using the substitution u = x² + 5.

2. Fluency

   Q2 — Substitution: trigonometric

   Find ∫sin³(x)cos(x) dx using u = sin x.

3. Understanding

   Q3 — Substitution: logarithmic form

   Find ∫ (2x) / (x² + 3) dx.

4. Understanding

   Q4 — Substitution: definite integral with changed limits

   Evaluate ∫₁³ x²√(x³ + 1) dx. Change the limits of integration.

5. Fluency

   Q5 — Integration by Parts: polynomial × exponential

   Find ∫xe^x dx using integration by parts.

6. Understanding

   Q6 — Integration by Parts: polynomial × logarithm

   Find ∫x ln x dx.

7. Problem Solving

   Q7 — Integration by Parts: repeated application

   Find ∫x²e^x dx using integration by parts twice.

8. Fluency

   Q8 — Partial Fractions: distinct linear factors

   Find ∫ 5 / [(x + 1)(x − 4)] dx by first expressing as partial fractions.

9. Understanding

   Q9 — Partial Fractions: repeated linear factor

   Find ∫ (3x + 1) / [x²(x − 1)] dx.

10. Understanding

    Q10 — Partial Fractions: irreducible quadratic factor

    Find ∫ (x + 3) / [(x² + 1)(x − 2)] dx.

11. Fluency

    Q11 — Trig Identities: even power reduction

    Find ∫sin²(x) dx using the identity sin²(x) = (1 − cos 2x)/2.

12. Understanding

    Q12 — Trig Identities: odd power with substitution

    Find ∫cos³(x) dx. Hint: write cos³(x) = cos²(x)cos(x) = (1 − sin²(x))cos(x).

13. Understanding

    Q13 — Trig Identities: product-to-sum formula

    Find ∫sin(3x)cos(x) dx using the product-to-sum identity: sin A cos B = (1/2)[sin(A+B) + sin(A−B)].

14. Problem Solving

    Q14 — Mixed: choose the right technique

    Find ∫ (x² + x + 1) / (x(x + 1)) dx. Determine whether partial fractions or substitution is more efficient.

15. Problem Solving

    Q15 — Mixed: trig integral requiring substitution and identity

    Evaluate ∫₀^π/4 sin(2x) / (1 + cos²(x)) dx. Hint: use sin(2x) = 2 sin x cos x and substitute u = cos x.