2023TrialShort answer

Q12 (b) (i)

3 marks

Question

Show that

\int_0^{\pi/2}\frac{dx}{1+\cos x+\sin x}=\ln 2.

Try the question before revealing the answer.

Solution

(b)(i)

Let $t=\tan\frac{x}{2}$ .

\frac{dt}{dx}=\frac12\sec^2\frac{x}{2}

\frac{dx}{dt}=\frac{2}{\sec^2\frac{x}{2}}=2\cos^2\frac{x}{2}=\frac{2}{1+t^2}

dx=\frac{2\,dt}{1+t^2}

When $x=\frac\pi2$ , $t=1$ .

When $x=0$ , $t=0$ .

\begin{aligned} \int_0^{\pi/2}\frac{dx}{1+\cos x+\sin x} &=\int_0^1\frac{\frac{2\,dt}{1+t^2}}{1+\frac{1-t^2}{1+t^2}+\frac{2t}{1+t^2}}\\ &=\int_0^1\frac{2\,dt}{1+t^2+1-t^2+2t}\\ &=\int_0^1\frac{2\,dt}{2+2t}\\ &=\int_0^1\frac{dt}{1+t}\\ &=\left[\ln|1+t|\right]_0^1\\ &=\ln|1+1|-\ln|1+0|\\ &=\ln2-0\\ &=\ln2 \end{aligned}

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Calculus

Further IntegrationApply substitution, partial fractions, integration by parts and recurrence relationships to theoretical and practical problems.

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