Question

By using the properties of definite integrals, evaluate the integral: $∫^{\frac{π}{2}}_0\frac{cos^5xdx}{sin^5x+cos^5x}$

Answer 1

The correct answer is:$I=\frac{π}{4}$
Let $I=∫^{\frac{π}{2}}_0\frac{cos^5xdx}{sin^5x+cos^5x}......(1)$
$⇒I=∫^{\frac{π}{2}}_0\frac{cos^5(\frac{π}{2}-x)}{sin^5(\frac{π}{2}-x)+cos^5(\frac{π}{2}-x)}dx\,\,\,\, (∫^a_0ƒ(x)dx=∫^a_0ƒ(a-x)dx)$
$⇒I=∫^{\frac{π}{2}}_0\frac{sin^5x}{sin^5x+cos^5x}dx...(2)$
Adding(1)and(2),we obtain
$2I=∫^{\frac{π}{2}}_0\frac{sin^5x+cos^5x}{sin^5x+cos^5x}dx$
$⇒2I=∫^{\frac{π}{2}}_01.dx$
$⇒2I=[x]^{\frac{π}{2}}_0$
$⇒2I=\frac{π}{2}$
$⇒I=\frac{π}{4}$