use the substitution u=tan(x) to evaluate the following?

Question

int tan-squared-x + sec^4x

the top limit is pi/4 and the bottom limit is 0.

any help would be great!

Answer 1

∫tan^2(x) + sec^4(x) dx, x from 0 to pi/4
= ∫[tan^2(x) + sec^4(x)][1/sec^2(x)] dtan(x), x from 0 to pi/4
= ∫[tan^2(x)/[1+tan^2(x)] + 1 + tan^2(x)] dtan(x), x from 0 to pi/4

Can you finish it now?

Answer 2

I = ∫ tan²x + sec^(4)x dx
I = ∫ tan²x + sec²x.sec²x dx
I = ∫ tan²x + (tan²x + 1).(tan²x + 1).dx
I = ∫ tan^(4)x + 3tan²x + 1 dx
let u = tan x
du = sec²x dx
dx = du / sec²x
dx = du / (tan²x + 1)
dx = du/(u² + 1)
I = ∫ (u^4 + 3u² + 1) / (u² + 1) du
I = ∫ u² + 2 - 1 / (u² + 1) du
I = u³/3 + 2u - tan^(-1) u + C
I = (1/3).tan³x + 2 tanx - tan^(-1)(tan x) + C