Indefinite Integral

Question

1. ∫(1+sec² x) / (1 + tan³ x)cos²x dx
2. ∫ cos x / sinx+sin³x dx

Answer 1

Use this graph to confirm the answer suggested in the video.

desmos.com/calculator/ugzxjrunhs

Note: only question 2 is answered in the video.

For part 1:

Replace (1+sec²x) with tan²x. Replace the cos²x in the denominator with sec²x in the numerator.

Let u = tanx, so du = sec²xdx.

At this point: ∫ u²du/(1+u³).

Now let w = 1 + u³, so dw = 3u²du

1/3∫ dw/w = 1/3 ln|w| = 1/3 ln|1+u³| = 1/3 ln|1 + tan³x| + C

desmos.com/calculator/zk4nkbyocp