Question

`int_0^(pi/4) (sec^2 x)/((1 + tan x)(2 + tan x))`dx = ?

Options

• log`(8/3)`

• log`(4/3)`

• `1/2 log (8/3)`

• `1/2 log (4/3)`

Answer 1

log`(4/3)`

Explanation:

Let I = `int_0^(pi/4) (sec^2 x)/((1 + tan x)(2 + tan x))`dx

Put tan x = t ⇒ sec²x dx = dt

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

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

`= [log |1 + "t"|]_0^1 - [log |2 + "t"|]_0^1`

= (log 2 - log 1) - (log 3 - log 2)

= log 2 - log`(3/2)`

∴ I = log `(4/3)`