Question

Evaluate the following : `int_0^(pi/4) (tan^3x)/(1 +cos2x)*dx`

Answer 1

Let I = `int_0^(pi/4) (tan^3x)/(1 +cos2x)*dx`

= `int_0^(pi/4) (tan^3x)/(2cos^2x)*dx`

= `(1)/(2) int_0^(pi/4) tan^3x*sec^2x*dx`

Put tan x = t
∴ sec²x·dx = dt
When x = 0, t = tan 0 = 0

When x = `pi/(4), t = tan  pi/(4)` = 1

∴ I = `(1)/(2) int_0^1 t^3*dt`

= `(1)/(2)*[(t^4)/4]_0^1`

= `(1)/(8)[t^4]_0^1`

= `(1)/(8)[1 - 0]`

= `(1)/(8)`.