Question

By using the properties of definite integrals, evaluate the integral: \(∫^\frac{π}{2}_0\frac{sinx-cosx}{1+sinx\,cosx\,}dx\)

Answer 1

Let I=\(∫^\frac{π}{2}_0\frac{sinx-cosx}{1+sinxcosxd}x........(1)\)

\(⇒I=∫^\frac{π}{2}_0\frac{sin(\frac{π}{2}-x)-cos(\frac{π}{2}-x)}{1+sin(\frac{π}{2}-x)cos(\frac{π}{2}-x)dx (∫a0ƒ(x)dx=∫a0ƒx)}dx)\)

\(⇒I=∫_0^{π}{2}\frac{cosx-sinx}{1+sinxcosx}dx...(2)\)

\(Adding(1)and(2),we obtain\)

\(2I=∫_0^\frac{π}{2}\frac{0}{1+sinxcosx}dx\)

\(⇒I=0\)