Definite Integration
Trigonometric Substitution
Grade None
Question:
<p>Evaluate \(\displaystyle\int_0^{\pi/2}\frac{\sin x}{\sin x+\cos x}\,dx\) [JEE Main 2016]</p>
Step-by-Step Solution
Key Concept: Let I = \int sinx/(sinx+cosx)dx, J = \int cosx/(sinx+cosx)dx. Then I+J = \pi/2 and I=J by King's rule.
<div class='solution'>
<p>Let $I=\int_0^{\pi/2}\frac{\sin x}{\sin x+\cos x}dx$. King ($x\to\pi/2-x$):</p>
<p>$$I=\int_0^{\pi/2}\frac{\cos x}{\cos x+\sin x}dx$$</p>
<p>Add: $2I=\int_0^{\pi/2}\frac{\sin x+\cos x}{\sin x+\cos x}dx=\int_0^{\pi/2}1\,dx=\frac{\pi}{2}\Rightarrow I=\boxed{\frac{\pi}{4}}$</p>
</div>
Correct Answer: A