Question

By using the properties of definite integrals, evaluate the integral: $∫^\frac{π}{2}_0(2logsinx-logsin2x)dx$

Answer 1

Let I=$∫^\frac{π}{2}_0(2logsinx-logsin2x)dx$

$⇒I=∫^\frac{π}{2}_0{{2log sinx-log(2sinx cosx)}}dx$

$⇒I=∫^\frac{π}{2}_0{2log sinx-logsinx-logcosx-log2}dx$

$⇒I=∫^{π}{2}_0{{logsinx-logcosx-log2}}dx...(1)$

$It is known that,(∫^a_0ƒ(x)dx=∫^a_0ƒ(a-x)dx)$

$⇒I=∫\frac{π}{2}_0{logcosx-logsinx-log2}dx...(2)$

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

$2I=∫^{π}{2}_0(-log2-log2)dx$

$⇒2I=-2log2∫^{π}{2}_01.dx$

$⇒I=-log2[\frac{π}{2}]$

$⇒I=\frac{π}{2}(-log2)$

$⇒I=\frac{π}{2}[log\frac{1}{2}]$

$⇒I=\frac{π}{2}log\frac{1}{2}$