Use the substitution $x=\cos heta $ to find the exact value of $\displaystyle\int_{\frac{\pi }{2}}

Question

Use the substitution $x=\cos 	heta $ to find the exact value of $\displaystyle\int_{\frac{\pi }{2}}^{\pi }{\frac{\sin 	heta }{1+{{\cos }^{2}}	heta }}\,\,\mathrm{d}	heta $. [5] Find $\displaystyle\int{{{\mathrm{e}}^{x}}\sin 2x\,\mathrm{d}x}$. [4] Hence find the exact value of $5\displaystyle\int_{0}^{\frac{\pi }{2}}{{{\mathrm{e}}^{x\,}}\sin \,2x\,\mathrm{d}x}$. [2]

Accepted Answer

$\frac{\pi}{4}$; $2\left(e^{\frac{\pi}{2}} + 1\right)$

2010 MI P1 Q11

2010 MI P1 Q11

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