CAIE A-Level Further Mathematics 2.4.1 Integration Substitutions

Practise recognising inverse trigonometric or hyperbolic integral forms and carrying out a stated substitution, including completing the square before applying exact limits.

Exam points

complete the square or rewrite the radical to identify the relevant inverse-function form
differentiate the stated substitution and transform both the integrand and limits consistently
return an indefinite integral to the original variable and retain the integration constant

[Maximum number: 2]

The integral $I_{n}$ , where n is an integer, is defined by $I_{n}=\int_{0}^{\frac{4}{3}}\left(1+x^{2}\right)^{\frac{1}{2} n} \mathrm{~d} x$ .

Find the exact value of $I_{-1}$ giving your answer in the form $\ln a$ , where a is an integer to be determined.

$I_{-1}=\int_{0}^{\frac{4}{3}}\left(1+x^{2}\right)^{-\frac{1}{2}} \mathrm{~d} x=\left[\sinh ^{-1} x\right]_{0}^{\frac{4}{3}}=\ln 3$

M1 A1

Recognises integral.

2

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2.4.1—Hyperbolic functions question 1