`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)" "xsec^(2)xdx`
`implies l=[xintsec^(2)xdx]_(0)^(pi//4)-int_(0)^(pi//4)((dx)/(dx)intsec^(2)xdx)dx` [integration by parts]
`implies l=[xtanx]_(0)^(pi//4)-int_(0)^(pi//4)tanxdx`
`implies l =[(pi)/(4)"tan"(pi)/(4)-0]-[logsecx]_(0)^(pi//4)`
`implies l=(pi)/(4)-("log sec"(pi)/(4)-"log sec 0")`
`l=(pi)/(4)-logsqrt(2)-log1`
`impliesl=(pi)/(4)-logsqrt(2)`