Evaluate $\displaystyle\int_{\pi}^{\frac{3\pi}{2}}\frac{\cos x}{3+\cos^{2}x}\,dx$ , using the substitution $u=\sin x$ . Give the exact answer as a single natural logarithm.

Model answer

1/4 ln(1/3)

Also accepted: (1/4)ln(1/3), $\frac{1}{4}\ln\frac{1}{3}$ , 0,25 ln(1/3), -1/4 ln 3, -(1/4)ln3, $-\frac{1}{4}\ln 3$

Explanation

With

u=\sin x

\cos x\,dx=du

and

3+\cos^{2}x=3+(1-u^{2})=4-u^{2}

. The limits become

u=\sin\pi=0

and

u=\sin\frac{3\pi}{2}=-1

, so the integral is

\int_{0}^{-1}\frac{du}{4-u^{2}}=\frac{1}{4}\left[\ln\frac{2+u}{2-u}\right]_{0}^{-1}=\frac{1}{4}\left(\ln\frac{1}{3}-\ln1\right)=\frac{1}{4}\ln\frac{1}{3}

Derived from ZIMSEC Mathematics Paper 2, June 2016, Q1

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