Evaluate: 1+ 5 x d x - Vidyadip

Question

Evaluate:

$\int \sqrt{1+\sin 5 x} \cdot d x$

✓

Answer

$\begin{aligned} & \int \sqrt{1+\sin 5 x} \cdot d x \\ = & \int \sqrt{\sin ^2 x+\cos ^2 x+5 \sin x \cos x} \cdot d x \\ = & \int \sqrt{(\sin x+\cos x)^2} \cdot d x \\ = & \int(\sin x+\cos x) \cdot d x \\ = & \int \sin x d x+\int \cos x \cdot d x \\ = & \left(\frac{2}{5} \sin \frac{5 x}{2}-\cos \frac{5 x}{2}\right)+c .\end{aligned}$

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