Integration (Method Selection) questions

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Integration (Method Selection) question collection

Review Integration (Method Selection) questions for Maths, with correct answers shown and coverage across choose the best method for \int \cos^2 x\,dx; choose the best method for \int \frac{1}{x^2-9}\,dx; choose the best method for \int \frac{3x^2}{x^3+4}\,dx.

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Question 1

Choose the best method for \(\int \cos^2 x\,dx\).
  1. \(\text{Trig identity}\)
  2. \(\text{Substitution: }u=\cos x\)
  3. \(\text{By parts}\)
  4. \(\text{Partial fractions}\)

Question 2

Choose the best method for \(\int \frac{1}{x^2-9}\,dx\).
  1. \(\text{Partial fractions}\)
  2. \(\text{Substitution: }u=x^2-9\)
  3. \(\text{By parts}\)
  4. \(\text{Reverse chain rule}\)

Question 3

Choose the best method for \(\int \frac{3x^2}{x^3+4}\,dx\).
  1. \(\text{Reverse chain rule / substitution}\)
  2. \(\text{Substitution: }u=3x^2\)
  3. \(\text{By parts}\)
  4. \(\text{Partial fractions}\)

Question 4

Choose the best method for \(\int \frac{4x}{(2x^2+1)^3}\,dx\).
  1. \(\text{Reverse chain rule}\)
  2. \(\text{Substitution: }u=4x\)
  3. \(\text{Partial fractions}\)
  4. \(\text{By parts}\)

Question 5

Choose the best method for \(\int \frac{5x+7}{(x+1)(x-2)}\,dx\).
  1. \(\text{Partial fractions}\)
  2. \(\text{Substitution: }u=(x+1)(x-2)\)
  3. \(\text{By parts}\)
  4. \(\text{Trig identity}\)

Question 6

Choose the best method for \(\int \operatorname{cosec} x\,dx\).
  1. \(\text{Formula booklet}\)
  2. \(\text{By parts}\)
  3. \(\text{Trig identity}\)
  4. \(\text{Substitution: }u=\operatorname{cosec} x\)

Question 7

Choose the best method for \(\int \sec x\,dx\).
  1. \(\text{Formula booklet}\)
  2. \(\text{By parts}\)
  3. \(\text{Trig identity}\)
  4. \(\text{Substitution: }u=\sec x\)

Question 8

Choose the best method for \(\int \sin^2 x\,dx\).
  1. \(\text{Trig identity}\)
  2. \(\text{Substitution: }u=\sin x\)
  3. \(\text{By parts}\)
  4. \(\text{Reverse chain rule}\)

Question 9

Choose the best method for \(\int 2x\cos(x^2+1)\,dx\).
  1. \(\text{Substitution: }u=x^2+1\)
  2. \(\text{Substitution: }u=2x\)
  3. \(\text{Substitution: }u=x^2-1\)
  4. \(\text{By parts}\)

Question 10

Choose the best method for \(\int 6x(3x^2+5)^4\,dx\).
  1. \(\text{Reverse chain rule}\)
  2. \(\text{Substitution: }u=6x\)
  3. \(\text{By parts}\)
  4. \(\text{Partial fractions}\)

Question 11

Choose the best method for \(\int x e^x\,dx\).
  1. \(\text{By parts: }u=x,\;v'=e^x\)
  2. \(\text{By parts: }u=e^x,\;v'=x\)
  3. \(\text{Substitution: }u=xe^x\)
  4. \(\text{Partial fractions}\)

Question 12

Choose the best method for \(\int x\sin x\,dx\).
  1. \(\text{By parts: }u=x,\;v'=\sin x\)
  2. \(\text{By parts: }u=\sin x,\;v'=x\)
  3. \(\text{Substitution: }u=x\sin x\)
  4. \(\text{Trig identity}\)

Question 13

Choose the best method for \(\int (3x^2-4x+1)\,dx\).
  1. \(\text{Integrate powers term by term}\)
  2. \(\text{Partial fractions}\)
  3. \(\text{By parts}\)
  4. \(\text{Trig identity}\)

Question 14

Choose the best method for \(\int \frac{1}{x}\,dx\).
  1. \(\text{Use }\ln|x|\)
  2. \(\text{Partial fractions}\)
  3. \(\text{By parts}\)
  4. \(\text{Trig identity}\)

Question 15

Choose the best method for \(\int e^{3x}\,dx\).
  1. \(\text{Reverse chain rule / substitution}\)
  2. \(\text{By parts}\)
  3. \(\text{Partial fractions}\)
  4. \(\text{Trig identity}\)

Question 16

Choose the best method for \(\int \cos(5x)\,dx\).
  1. \(\text{Reverse chain rule}\)
  2. \(\text{Partial fractions}\)
  3. \(\text{By parts}\)
  4. \(\text{Trig identity}\)

Question 17

Choose the best method for \(\int \frac{1}{2x+1}\,dx\).
  1. \(\text{Reverse chain rule / substitution}\)
  2. \(\text{By parts}\)
  3. \(\text{Partial fractions}\)
  4. \(\text{Trig identity}\)

Question 18

Choose the best method for \(\int \frac{x}{x^2+4}\,dx\).
  1. \(\text{Reverse chain rule / substitution}\)
  2. \(\text{By parts}\)
  3. \(\text{Partial fractions}\)
  4. \(\text{Trig identity}\)

Question 19

Choose the best method for \(\int \ln x\,dx\).
  1. \(\text{By parts: }u=\ln x,\ v'=1\)
  2. \(\text{Partial fractions}\)
  3. \(\text{Trig identity}\)
  4. \(\text{Substitution: }u=\ln x\)

Question 20

Choose the best method for \(\int x^2\ln x\,dx\).
  1. \(\text{By parts: }u=\ln x,\ v'=x^2\)
  2. \(\text{Partial fractions}\)
  3. \(\text{Trig identity}\)
  4. \(\text{Substitution: }u=x^2\ln x\)

Question 21

Choose the best method for \(\int \frac{1}{x^2+4}\,dx\).
  1. \(\text{Standard arctangent result}\)
  2. \(\text{Partial fractions}\)
  3. \(\text{Integration by parts}\)
  4. \(\text{Trigonometric identity}\)

Question 22

Choose the best method for \(\int \frac{2x+3}{x(x+1)}\,dx\).
  1. \(\text{Partial fractions}\)
  2. \(\text{By parts}\)
  3. \(\text{Trig identity}\)
  4. \(\text{Reverse chain rule}\)

Question 23

Choose the best method for \(\int e^x\sin x\,dx\).
  1. \(\text{Repeated integration by parts}\)
  2. \(\text{Partial fractions}\)
  3. \(\text{Trig identity}\)
  4. \(\text{Substitution: }u=e^x\)

Question 24

Which expression equals \(\tan x\)?
  1. \(\frac{\sin x}{\cos x}\)
  2. \(\frac{\cos x}{\sin x}\)
  3. \(\sin x\cos x\)
  4. \(\sin x+\cos x\)

Question 25

Choose the best method for \(\int \sin x\cos x\,dx\).
  1. \(\text{Substitution: }u=\sin x\)
  2. \(\text{Partial fractions}\)
  3. \(\text{By parts}\)
  4. \(\text{Formula booklet}\)

Question 26

Complete \(\cos^3x=\cos x(\ldots)\) before integrating.
  1. \(1-\sin^2x\)
  2. \(1-\cos^2x\)
  3. \(1+\sin^2x\)
  4. \(1+\cos^2x\)

Question 27

Choose the best method for \(\int \frac{3}{(1+x)^2}\,dx\).
  1. \(\text{Reverse chain rule / substitution}\)
  2. \(\text{Partial fractions}\)
  3. \(\text{By parts}\)
  4. \(\text{Trig identity}\)

Question 28

Choose the best method for \(\int \frac{2x+1}{\sqrt{x^2+x+5}}\,dx\).
  1. \(\text{Reverse chain rule}\)
  2. \(\text{Partial fractions}\)
  3. \(\text{By parts}\)
  4. \(\text{Trig identity}\)

Question 29

Choose the best method for \(\int \frac{1}{x(x+1)}\,dx\).
  1. \(\text{Partial fractions}\)
  2. \(\text{By parts}\)
  3. \(\text{Trig identity}\)
  4. \(\text{Reverse chain rule}\)

Question 30

Choose the best method for \(\int x\sqrt{x+1}\,dx\).
  1. \(\text{Substitution: }u=x+1\)
  2. \(\text{Partial fractions}\)
  3. \(\text{Trig identity}\)
  4. \(\text{Formula booklet}\)

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Review Integration (Method Selection) questions for Maths, with correct answers shown and coverage across choose the best method for \int \cos^2 x\,dx; choose the best method for \int \frac{1}{x^2-9}\,dx; choose the best method for \int \frac{3x^2}{x^3+4}\,dx.

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