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

ridge041h

Answered question

2022-09-14

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?

Answer & Explanation

Waylon Jenkins

Waylon Jenkins

Beginner2022-09-15Added 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

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Equations

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?