Mathematics

# Evaluate:$\displaystyle\int\limits_0^{\dfrac{\pi }{2}} {\dfrac{{\sin x}}{{\sin x + \cos x}}dx}$

##### SOLUTION

Consider the given integral.

$I=\int_{0}^{\dfrac{\pi }{2}}{\left( \dfrac{\sin x}{\sin x+\cos x} \right)}dx$    …… (1)

We know that

$\int_{a}^{b}{f\left( x \right)}dx=\int_{a}^{b}{f\left( a+b-x \right)}dx$

Therefore,

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

$I=\int_{0}^{\dfrac{\pi }{2}}{\left( \dfrac{\cos x}{\cos x+\sin x} \right)}dx$      ….. (2)

From equation (1) and (2), we get

$2I=\int_{0}^{\dfrac{\pi }{2}}{\left( \dfrac{\cos x+\sin x}{\cos x+\sin x} \right)}dx$

$2I=\int_{0}^{\dfrac{\pi }{2}}{1}dx$

$2I=\left( x \right)_{0}^{\dfrac{\pi }{2}}$

$2I=\left( \dfrac{\pi }{2}-0 \right)$

$I=\dfrac{\pi }{4}$

Hence, the value is $\dfrac{\pi }{4}$.

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
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