The value of the integral `int((1-costheta)^(2/7))/((1+costheta)^(9/7))d theta` is

Options

A. `(7)/(11)("tan"(theta)/(2))^((11)/(7))+C`
B. `(7)/(11)("cos"(theta)/(2))^((11)/(7))+C`
C. `(7)/(11)("sin"(theta)/(2))^((11)/(7))+C`
D. none of these

Answer

A

Explanation

Let `I=int((1-cos theta)^(2//7))/((1+cos theta)^(9//7))d theta`
`=int((2sin^(2)theta//2)^(2//7))/((2cos^(2)theta//2)^(9//7))d theta`
`=(1)/(2)int((sin theta//2)^(4//7))/((cos theta//2)^(18//7)) d theta`
Put `(theta)/(2)=t" or " (d theta)/(2)=dt`
` :. I=int((sint)^(4//7))/((cost)^(18//7)) dt " "("Here, "m+n= -2)`
`=int(tant)^(4//7)sec^(2)t dt`
Put `tan t=u" or "sec^(2)t dt=du`
` :. I=int u^(4//7)du=(u^(11//7))/(11//7)+c=(7)/(11)(tan t)^(11//7)+C`
`=(7)/(11)("tan"(theta)/(2))^(11//7)+C`