Evaluate each integral

Question

∫ sec^2(2𝜃) 𝑑𝜃 Evaluate each integral

Answer 1

The derivative of tan is sec^2.

Put y = 2θ and dy = 2 dθ

∫sec²(2θ) dθ = ∫sec²y (1/2) dy = (1/2) tan y + C = (1/2) tan 2θ + C

Answer 2

Let u = 2θ

Then du/dθ = 2 or, rearranging, dθ = du/2

Using ∫sec²(2θ) dθ use the substitution above to re-write the integral:

∫sec²(u) du/2 or, bringing the 1/2 out in front: 1/2∫sec²(u) du

Since the derivative of tan(u) = sec²(u), then the antiderivative of sec²(u) is tan(u) so the integral becomes:

1/2 tan(u) + C

Now, replace the u with 2θ making the final answer:

1/2 tan(2θ) + C