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

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

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

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

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

$⇒I=0$