Question

Evaluate the following indefinite integrals.

Answer 1

^ oh yeah u substitution :D

Answer 2

@Albertoimus Do you get it? I know it looks a little messy but if you want I can rewrite so you can see it better.

Answer 3

by all means.

Answer 4

\[\int\limits_{}^{}\frac{\cos \sqrt{x}}{\sqrt{x}}dx\]

where we say that
\[u=\sqrt{x}\]and if we square both sides that means that
\[x=u^2\]and if we take the derivative of that we get
\[dx=2u\]

Answer 5

The next step is a matter of replacing things

Answer 6

woops the last part should be \[dx=2udu\]

Answer 7

\[\int\limits_{}^{}\frac{cosu}{u}*2udu\]
we cancel the u on the top with the u on the bottom.
Also since 2 is a constant we take it outside the integral \[2\int\limits_{}^{}cosu*du\]
the integral of cos u du is simply sin u
\[2* \sin u\]

And now we want it to convert it back into x instead of u and we know already that
\[u=\sqrt{x}\]

So it comes out as \[2*\sin \sqrt{x}\]

Answer 8

Done!

Answer 9

thanks