Integration by Parts: Evaluating e^xcos3x dx & x^2/(2(1+x^2))

[Illusionist]

Homework Statement

Hi, I'm trying to evaluate the integration of (e^x)cos3x dx, by using integration by parts. I've already done a couple of similar questions on integration by parts but this one seems to puzzle me.

Homework Equations

The answer is supposed to be (e^x/10)*(cos3x + 3sin3x), I can't seem to get to it.

The Attempt at a Solution

I've basically taken the same approach to this question as the other couple of integration by parts questions I've done but the answer is never right. I let u=cos3x and dv/dx=e^x. And basically just subbed back into the formula uv - integ.v*(du/dx).
I suspect I may be anti-differentiating the second part wrong, which I think is (e^x)*3sin3x. I get 3e^x(3cos3x+sin3x). Any idea where I went wrong?

I'm also having trouble finding the integral of (x^2)/[2(1+x^2)]. Any advice or help would be appreciated. Thanks in advance.

 

[D H]

Integration by parts is so much fun tht you should do it twice.

At least in this case you should.

 

[Mathgician]

On the integration by parts after you integrate twice by parts look on the left and right side of the equation and see if you see anything common like:

x= y+ 3x
just imagine that x is the original integral, when you get like terms on the right side, you can manipulate the integral to get left equal right. I hope you can see what I am saying.

On the second integral, divide and split the integral and see if it would make it easier for you to integrate

 

[Gib Z]

Yup On the second one, Add 1 and minus 1 from the numerator. Don't think its stupid, just do it! And take the factor of 1/2 out of the integral.

 

[Illusionist]

Oh I see. I get everything now, thanks a bunch guys. Very appreciated.