Unit 5: Calculus

Topic 5.16: Integration by parts (HL only) Questions

Practice 20 exam-style questions for IB Math AA SL Topic 5.16. Review the question stems below, then unlock the full Question Bank to access markschemes, model answers, and AI grading.

11 mark
2026
For a definite integral, after substituting u you should:
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21 mark
2026
In LIATE, which factor should be u in ∫ x³ ln x dx?
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3Show that1 mark
2026
Show that by differentiating the right-hand side.
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41 mark
2026
Integration by parts comes from reversing which rule?
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51 mark
2026
To integrate ∫ 5(5x − 1)³ dx, the substitution u = 5x − 1 gives:
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6Evaluate3 marks
2026
Evaluate ∫₁e ln x dx.
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7Find3 marks
2026
Find ∫ 6x(3x² + 5)⁴ dx.
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81 mark
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∫ 1/(2x + 5) dx equals:
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91 mark
2026
After u = x² + 1 in ∫₀¹ 2x(x² + 1)⁵ dx, the integral becomes:
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10Find3 marks
2026
Find ∫ sin⁴x cos x dx.
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111 mark
2026
Which integral is set up for u = cos x (du = −sin x dx)?
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121 mark
2026
∫ x ex dx equals:
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13Find3 marks
2026
Find ∫ x sin x dx.
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141 mark
2026
If after one round of parts the new integral looks HARDER, you should:
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151 mark
2026
How many applications of parts does ∫ x³ ex dx need?
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16Find5 marks
2026
Find .
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17Find3 marks
2026
Find ∫ x e2x dx.
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18Find4 marks
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Find ∫ x² sin x dx.
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19Show that5 marks
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Show that .
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20Evaluate4 marks
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Evaluate ∫₀π/2 x cos x dx.
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