## Integral solving problem

1. The problem statement, all variables and given/known data

I've been trying to integrate the following: ∫$\frac{cos^3(x)}{\sqrt{sin(x)}}$dx

2. Relevant equations

3. The attempt at a solution

First, I substituted sin(x) with t, and got dt=cos(x)dx => dx=$\frac{dt}{cos(x)}$.
After that I got ∫$\frac{cos^2(x)}{\sqrt{t}}$dt
Then i transformed cos^2(x) into 1-sin^2(x), and finally got to ∫$\frac{1-t^2}{\sqrt{t}}$dt

I thought I could just disintegrate them into two smaller integrals like ∫$\frac{1}{\sqrt{t}}$dt - ∫$\frac{t^2}{\sqrt{t}}$dt , and solve them easily, and then reverse the substitution.

Wolfram proposes that i cannot(?) do that, or rather prefers that I do another substitution.

I even tried to make it a defined integral, and calculate the values between the Wolfram solution, and my own. They differ by 0.1 or something similar.

Can someone explain what is the right way to do it?

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 Recognitions: Homework Help Both are correct. Wolfram just does another substitution which isn't really too necessary. Both give the same correct integral.
 Thank you very much for your help!