Find \(\displaystyle\displaystyle \displaystyle\int{\frac{{{\mathrm{e}}^{x}}}{{{\left( 1+2{{\mathrm{

Question

Find $\displaystyle\displaystyle \displaystyle\int{\frac{{{\mathrm{e}}^{x}}}{{{\left( 1+2{{\mathrm{e}}^{x}} \right)}^{3}}}}\,\mathrm{d}x$[2] Find $\displaystyle\displaystyle \displaystyle\int{\frac{x-1}{1+4{{x}^{2}}}}\,\mathrm{d}x$[3] Find $\displaystyle\displaystyle \displaystyle\int{x\,{{(\ln x)}^{2}}\,\mathrm{d}x}$.[4]

Accepted Answer

(a) $-\frac1{4(1+2e^x)^2}+C$. (b) $\frac18\ln(1+4x^2)-\frac12\tan^{-1}(2x)+C$. (c) $\frac{x^2}{2}(\ln x)^2-\frac{x^2}{2}\ln x+\frac{x^2}{4}+C$, $x>0$.

2020 TMJC Promo Q5

2020 TMJC Promo Q5

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