Find $\displaystyle\displaystyle \displaystyle\int{\frac{1}{{{x}^{2}}+2x+5}\,\mathrm{d}x}$,[2] \(\

Question

Find $\displaystyle\displaystyle \displaystyle\int{\frac{1}{{{x}^{2}}+2x+5}\,\mathrm{d}x}$,[2] $\displaystyle\displaystyle \displaystyle\int{\frac{{{(\ln x)}^{3}}}{x}\,\mathrm{d}x}$.[2] Find the exact value of $\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{\pi }{2}}{x\cos x\,\mathrm{d}x}$.[3]

Accepted Answer

(a)(i) $\frac12\tan^{-1}((x+1)/2)+C$. (a)(ii) $\frac14(\ln x)^4+C$, $x>0$. (b) $\frac\pi2-1$.

2020 VJC Promo Q2

2020 VJC Promo Q2

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