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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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