CAIE A-Level Further Math A2 2.4.1 Hyperbolic Functions Questions

Practise choosing and applying substitutions to evaluate exact definite and indefinite integrals with Further Mathematics Paper 2 questions and mark schemes.

Exam points

Use substitutions to evaluate exact definite and indefinite integrals.

[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.

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

2

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CAIE A-Level Further Math A2 2.4.1 Hyperbolic Functions Questions question 1