Evaluate the definite integrals
`int_(0)^((pi)/(4))(sinx+cosx)/(9+16sin2x)dx`

Options

A. `1/20 log3`
B. `1/40 log3`
C. `1/20 log 6`
D. `10 log 3`

Answer

A

Explanation

`I=int_(0)^(pi//4)(sinx+cosx)/(25-16(sinx+cosx)^(2)) dx`
Let `sinx-cosx=t`
`:.I=int_(-1)^(0)(dt)/(25-16t^(2))`
`=1/16int_(-1)^(0)(dt)/((5/4)^(2)-t^(2))`
`=1/16 . 1/2 . 5/4 log [|(5/4+t)/(5/4-t)|]_(-1)^(0)`
`=1/40[log1-"log"1/9]`
`=(log 9)/40`
`=1/20 log 3`