Question

`int_0^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals ______.

Options

• `log_e  2/3`

• log_e3

• `1/2log_e  4/3`

• `log_e  4/3`

Answer 1

`int_0^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals `underlinebb(log_e  4/3)`.

Explanation:

Let tanx = t

⇒ sec²xdx = dt

At, `{{:(x = 0",", t = 0),(x = π/4",", t = 1):}`

I = `int_0^1 (dt)/((1 + t)(2 + t))`

= `int_0^1(1/(1 + t) - 1/(2 + t))dt`

= `[ℓn(1 + t) - ℓn(2 + t)]_0^1`

= ln2 – ln3 + ln2

= `ln  4/3`

= `log_e  4/3`