int_{0}^{1}cos(x^{2})dx=1/2 int_{0}^{1}cosx/sqrt(x) dx

podvelkaj8

podvelkaj8

Answered question

2022-10-18

Equality of two trigonometric integrals on [0,1]
I need to show, that:
0 1 cos ( x 2 )   d x = 1 2 0 1 cos x x   d x
But frankly I cannot see way to solve it. The right-side integral is improper and as far I know both of them don't have the elementary antiderivatives.

Answer & Explanation

Kamden Simmons

Kamden Simmons

Beginner2022-10-19Added 10 answers

Explanation:
Set t = x 2 . Then x = t , d x = 1 2 t d t.
Then change the bounds: if x = 0 , t = 0, if x = 1 , t = 1. So the two integrals are equal. The name of the variable does not chage the things.

Do you have a similar question?

Recalculate according to your conditions!

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?