Question

Evaluate the following integral:

\[\int_0^\frac{\pi}{2} \frac{a\sin x + b\sin x}{\sin x + \cos x}dx\]

 

Answer 1

\[\text{Let I }=\int_0^\frac{\pi}{2} \frac{a\sin x + b\sin x}{\sin x + \cos x}dx...............(1)\]

Then,

\[I = \int_0^\frac{\pi}{2} \frac{a\sin\left( \frac{\pi}{2} - x \right) + b\cos\left( \frac{\pi}{2} - x \right)}{\sin\left( \frac{\pi}{2} - x \right) + \cos\left( \frac{\pi}{2} - x \right)}dx ...................\left[ \int_0^a f\left( x \right)dx = \int_0^a f\left( a - x \right)dx \right]\]

\[= \int_0^\frac{\pi}{2} \frac{a\cos x + b\sin x}{\cos x + \sin x}dx................(2)\]

Adding (1) and (2), we get

\[2I = \int_0^\frac{\pi}{2} \left( \frac{a\sin x + b\cos x}{\cos x + \sin x} + \frac{a\cos x + b\sin x}{\sin x + \cos x} \right)dx\]
\[ \Rightarrow 2I = \int_0^\frac{\pi}{2} \left( \frac{a\sin x + b\cos x + a\cos x + b\sin x}{\sin x + \cos x} \right)dx\]
\[ \Rightarrow 2I = \int_0^\frac{\pi}{2} \frac{\left( a + b \right)\sin x + \left( a + b \right)\cos x}{\sin x + \cos x}dx\]
\[ \Rightarrow 2I = \int_0^\frac{\pi}{2} \frac{\left( a + b \right)\left( \sin x + \cos x \right)}{\sin x + \cos x}dx\]

\[\Rightarrow 2I = \int_0^\frac{\pi}{2} \left( a + b \right)dx\]
\[ \Rightarrow 2I = \left( a + b \right) \times \left.x\right|_0^\frac{\pi}{2} \]
\[ \Rightarrow 2I = \left( a + b \right) \times \left( \frac{\pi}{2} - 0 \right)\]
\[ \Rightarrow 2I = \frac{\pi}{2}\left( a + b \right)\]
\[ \Rightarrow I = \frac{\pi}{4}\left( a + b \right)\]