Mathematics

Evaluate the following definite integral:

$$\displaystyle\int_{0}^{\pi/2}\dfrac{\sin x+\cos x}{3+\sin 2x}dx$$


SOLUTION

Let $$I=\displaystyle\int_{0}^{\pi/4}\dfrac{\sin x+\cos x}{3+\sin 2x}\ dx$$ 

$$I=\displaystyle\int_{0}^{\pi/4}\dfrac{\sin x+\cos x}{4-(\sin x-\cos x)^{2}}\ dx$$

$$\Rightarrow I=\dfrac{1}{4}\left[\log\left|\dfrac{2+\sin x-\cos x}{2-\sin x+\cos x}\right|\right]_{0}^{\pi/4}$$ 

$$I=-\dfrac{1}{4}\log\dfrac{1}{3}=\dfrac{1}{4}\log_{e}3$$

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Subjective Medium Published on 17th 09, 2020
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