How did the answer found the antiderivative of (a)/(a^2+cos^2 x) to be (1)/(sqrt(1+a^2)) tan^(-1) ((a tan x)/(sqrt(1+a^2))) ?

ridge041h 2022-09-14 Answered
How did the answer found the antiderivative of a a 2 + cos 2 x to be 1 1 + a 2 tan 1 ( a tan x 1 + a 2 )?
I have not done Laplce transform previously and so is this purely by Laplace transform? I could verify that this is indeed the anti-derivative but could we get around Laplace transform to find the antiderivative straightaway?
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Answers (1)

Waylon Jenkins
Answered 2022-09-15 Author has 17 answers
When the integrand has only even powers of sin x and cos x, the substitution z = tan x can be useful:
α α 2 + cos 2 x d x = α d z α 2 + 1 + α 2 z 2 = d θ 1 + α 2 = θ 1 + α 2 + C = 1 1 + α 2 tan 1 α z 1 + α 2 + C = 1 1 + α 2 tan 1 α tan x 1 + α 2 + C
Where we have made the further substitution α z = 1 + α 2 tan θ

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