Evaluate the following : 1+ 3 x d x - Vidyadip

Question

Evaluate the following : $\int \sqrt{1+\sin 3 x} \cdot d x$

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Answer

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

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