I don't get those integrals

Please show me how do I solve these. I need the way.
(e^(1/x))/x  dx

cos(x) * 2^(sin x)   dx

(tan(5x))^5  dx

(sin(x/2))^3  *  (cos(x/2))^5   dx

(cos(3x))^4  dx

Thanks ahead and good night.

1) INTEGRATION BY PARTS: f(x) = e^(1/x), g'(x) = 1/x

Can't integrate the first one in closed form.

>>2 is wrong.

Second looks like it might be a simple reverse chain rule. See what happens if you differentiate 2^(sinx)

Those are quite nasty (by that I mean laborious), but they all seem solvable using things like Integration by parts, u-substitution (the "new" term for reversed chain rule.

First one: looking at it, it comprises of a composition and a product. Try u-substitution with either 1/x as u or e^{1/x} as u. I'm not bothered working it out that far. If u-substitution on its own fails, use IBP (int. by parts) and u substitution (but I would guess that would fail).

Second: what >>4 said.

Third: Try using IBP where u (the one to be differentiated) is (tan(5x))^5 and dv (the one to be integrated) is 1. Think of the trick used to integrate ln(x).

Fourth: Similar to the second.

Fifth: Similar to the third.

These are educated guesses, I haven't bothered working them out, so don't expect my postulates to bear fruit instantly without some thought from you.

>>4
d/dx a^f(x) = df/dx*a^f(x)*ln(a)

Off by a factor of ln(2).

It's too late now but thanks anyways.

If you need to check your answers
http://integrals.wolfram.com/index.jsp

>>8

Bookmarked.

Thanks for that!

Another one to try is "wolframalpha.com". It can even show the steps for integrals you'd encounter in an Elementary Calculus course.