Find $\displaystyle\displaystyle \displaystyle\int{\frac{\cos x}{\sqrt[3]{\sin x}}}\,\mathrm{d}x$.

Question

Find $\displaystyle\displaystyle \displaystyle\int{\frac{\cos x}{\sqrt[3]{\sin x}}}\,\mathrm{d}x$.[2] Using the substitution $x=1-{{\mathrm{e}}^{u}}$, find $\displaystyle\displaystyle \displaystyle\int{\frac{1}{1-{{\mathrm{e}}^{u}}}\,\mathrm{d}u}$.[4] Find $\displaystyle\displaystyle \displaystyle\int{x\ln \left( x+1 \right)\,\mathrm{d}x}$, where $x>-1$.[4] Write down constants $A$ and $B$ such that, for all values of $x$, $x=A\left( 2x+2 \right)+B$.[2] Hence, find $\displaystyle\displaystyle \displaystyle\int{\frac{x}{{{x}^{2}}+2x+2}\,\mathrm{d}x}$.[4]

Accepted Answer

(a) $\frac{3}{2}{{\left( \sin x \right)}^{\frac{2}{3}}}+C$ (b) $-\ln \left| 1-{{\text{e}}^{u}} \right|+u+C$ (c) $\frac{{{x}^{2}}}{2}\ln (x+1)-\frac{1}{2}\left[ \frac{{{x}^{2}}}{2}-x+\ln \left( x+1 \right) \right]+C$ (d)(i) $-1$ (ii) $\frac{1}{2}\ln \left( {{x}^{2}}+2x+2 \right)-{{\tan }^{-1}}\left( x+1 \right)+C$

2023 JPJC Promo Q13

2023 JPJC Promo Q13

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