Evaluate the following :
`int_(0)^(pi//4)x sec^(2)xdx`

Options

A. `(pi)/(4)+logsqrt(2)`
B. `(pi)/(4)-logsqrt(2)`
C. `1+logsqrt(2)`
D. `1-(1)/(2)log2`

Answer

B

Explanation

Let
`l=int_(0)^(pi//4) x sec^(2)x dx`
`rArr l=[x int sec^(2)x dx]_(0)^(pi//4)-int_(0)^(pi//4)((dx)/(dx) int sec^(2) x dx ) dx` [ integration by parts]
`rArr l=[x tan x]_(0)^(pi//4)- int_(0)^(pi//4) tan x dx `
`rArr l=[(pi)/(4) tan . (pi)/(4) - 0 ] - [ log sec x]_(0)^(pi//4)`
`rArr l=(pi)/(4)-(log sec . (pi)/(4)- log sec 0)`
`l=(pi)/(4)- log sqrt(2)-log 1`
`rArr l=(pi)/(4)- log sqrt(2)`