Question

`int_0^(pi4) sec^4x  "d"x` = ______.

Options

• `1/3`

• `4/3`

• `7/3`

• `8/3`

Answer 1

`int_0^(pi4) sec^4x  "d"x` = `4/3`.

Explanation:

Let I = `int_0^(pi/4) sec^4x  "d"x`

= `int_0^(pi/4) sec^2x.sec^2x  "d"x`

= `int_0^(pi/4) (1 + tan^2x)sec^2x  "d"x`

Put tan x = t

⇒ `sec^2x  "d"x` = dt

∴ I = `int_0^1 (1 + "t"^2)"dt"`

= `["t" + "t"^3/3]_0^1`

= `1 + 1/3 - 0`

= `4/3`