Question
Evaluate the following integrals:
$\int^\limits\frac{\pi}{4}_{0}\sin^32\text{t}\cos2\text{t}\text{ dt}$
$\int^\limits\frac{\pi}{4}_{0}\sin^32\text{t}\cos2\text{t}\text{ dt}$
Let
$\sin2\text{t}=\text{u}$ Then, $2\cos2\text{t dt} =\text{du}$When
$\text{t}=0,\text{u}=0$ and $\text{t}=\frac{\pi}{4},\text{u}=1$$\therefore\ \text{I}=\frac{1}{2}\int^\limits{1}_{0}\text{u}^3\text{ du}$
$\Rightarrow\text{I}=\frac{1}{2}\Big[\frac{\text{u}^4}{4}\Big]^1_0$
$\Rightarrow\text{I}=\frac{1}{2}\Big(\frac{1}{4}-0\Big)$
$\Rightarrow\text{I}=\frac{1}{8}$
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