Express $\frac{2x}{(x+1)(x^2+1)} = \frac{A}{x+1} + \frac{Bx+C}{x^2+1}$, where $A$, $B$ an

Question

Express $\frac{2x}{(x+1)(x^2+1)} = \frac{A}{x+1} + \frac{Bx+C}{x^2+1}$, where $A$, $B$ and $C$ are constants to be found.[1] Evaluate $\displaystyle\int_0^1 \frac{\ln(1+x^2)}{(x+1)^2} \, \mathrm{d}x$, giving your answer in the form $a\pi - \ln b$, where $a$ and $b$ are positive constants to be found.[7] Find $\displaystyle\int \sin^3(kx) \, \mathrm{d}x$ where $k$ is a constant.[3]

Accepted Answer

(a)(i) $A=-1,B=1,C=1$. (ii) $\frac{\pi}{4}-\ln2$, with $a=\frac{1}{4},b=2$. (b) For $k\ne0$, $\frac{1}{k}(-\cos(kx)+\frac{1}{3}\cos^3(kx))+C$; for $k=0$, $C$.

2024 DHS Promo Q11

2024 DHS Promo Q11

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