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Indefinite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integration2 Marks
Question
Evaluate the following : $\int \sin ^4 x \cdot d x$
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Answer

$
\begin{aligned}
I & =\int\left(\sin ^2 x\right)^2 \cdot d x \\
& =\int\left(\frac{1}{2}(1-\cos 2 x)\right)^2 \cdot d x \\
& =\frac{1}{4} \cdot \int\left(1-2 \cos 2 x+\cos ^2 2 x\right) \cdot d x \\
& =\frac{1}{4} \cdot \int\left[1-2 \cos 2 x+\frac{1}{2}(1+\cos 4 x)\right] \cdot d x \\
& =\frac{1}{4} \cdot \int\left(1-2 \cos 2 x+\frac{1}{2}+\frac{1}{2} \cos 4 x\right) \cdot d x \\
& =\frac{1}{4} \cdot \int\left(\frac{3}{2}-2 \cos 2 x+\frac{1}{2} \cos 4 x\right) \cdot d x \\
& =\frac{1}{4} \cdot\left[\frac{3}{2} x-2 \sin 2 x \cdot \frac{1}{2}+\frac{1}{2} \sin 4 x \cdot \frac{1}{4}\right]+c \\
& =\frac{1}{4} \cdot\left[\frac{3}{2} x-\sin 2 x+\frac{1}{8} \sin 4 x\right]+c
\end{aligned}
$

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