Question

`int_0^(π/4) x. sec^2 x  dx` = ______.

Options

• `π/4 + log sqrt(2)`

• `π/4 - log sqrt(2)`

• `1 + logsqrt(2)`

• `1 - 1/2 log 2`

Answer 1

`int_0^(π/4) x. sec^2 x  dx` = `underlinebb(π/4 - log sqrt(2)`.

Explanation:

`int_0^(π/4) x sec^2 x  dx`

= `[x int sec^2 x  dx]_0^(π/4) - int_0^(π/4)[d/dx x int sec^2 x  dx]dx`

= `[x. tan x]_0^(π/4) - int_0^(π/4) [tan x]dx`

= `[x.tan x]_0^(π/4) - [log |sec x|]_0^(π/4)`

= `[π/4 - 0] - [log|sec  π/4| - log |sec 0|]`

= `π/4 - [log sqrt(2) - log 1]`

= `π/4 - log sqrt(2)`