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) $-\frac{1}{x+1}+\frac{x+1}{{{x}^{2}}+1}$ (a)(ii) $=\frac{\pi }{4}-\ln 2$ (b) $\frac{1}{k}\left( -\cos kx+\frac{1}{3}{{\cos }^{3}}\left( kx \right) \right)+c$

2024 DHS Promo Q11

2024 DHS Promo Q11