Let I=\(∫^\frac{π}{2}_0(2logsinx-logsin2x)dx\)
\(⇒I=∫^\frac{π}{2}_0{{2log sinx-log(2sinx cosx)}}dx\)
\(⇒I=∫^\frac{π}{2}_0{2log sinx-logsinx-logcosx-log2}dx\)
\(⇒I=∫^{π}{2}_0{{logsinx-logcosx-log2}}dx...(1)\)
\(It is known that,(∫^a_0ƒ(x)dx=∫^a_0ƒ(a-x)dx)\)
\(⇒I=∫\frac{π}{2}_0{logcosx-logsinx-log2}dx...(2)\)
\(Adding(1)and(2),we obtain\)
\(2I=∫^{π}{2}_0(-log2-log2)dx\)
\(⇒2I=-2log2∫^{π}{2}_01.dx\)
\(⇒I=-log2[\frac{π}{2}]\)
\(⇒I=\frac{π}{2}(-log2)\)
\(⇒I=\frac{π}{2}[log\frac{1}{2}]\)
\(⇒I=\frac{π}{2}log\frac{1}{2}\)
Evaluate:
$\displaystyle \int_{0}^{3} x \cos(\pi x) \, dx$
Find:$\displaystyle \int \dfrac{dx}{\sin x + \sin 2x}$