Evaluate: _0^ 4 2x 3x dx - Vidyadip

Question

Evaluate:

$\int\limits_0^\frac{\pi}{4} \sin 2x \sin 3\text{x dx}$

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Answer

$\frac{1}{2}\int\limits_0^\frac{\pi}{4} 2\sin 2x \sin 3\text{x dx}$

$= \frac{1}{2} \int\limits_0^\frac{\pi}{4} (\cos x - \cos 5\text{x)dx}$

$= \frac{1}{2} \bigg [\sin x - \frac{\sin 5x}{5}\bigg]_{0}^{\frac{\pi}{4}}$

$= \frac{1}{2} \bigg[\sqrt\frac{1}{2} + \frac{1}{5} \sqrt\frac{1}{2}\bigg] = \frac{1}{2} \bigg[\frac{5 + 1}{5\sqrt{2}}\bigg] = \frac{3}{5\sqrt{2}} $

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