How Do You Solve This Integral Using Partial Fractions and U-Substitution?

[Nick20]

Hey! Need some help on an integral I've been trying to solve. I haven't been working enough with my math, so I am working on catching up before finals..

I was working with partial fractions and u substitution when I couldn't figure this one out:

∫(x+5)/(x^2+4x+8)dx

it's not that hard but I am missing one small detail in order to solve it, I am sure.

I believe the answer has something to do with

∫1/(a^2+x^2)dx = 1/a arctan(x/a) + C

Thanks

 

[arildno]

1. Complete the square in the denominator!
2. In the numerator, write 5=2+3, and then see if an idea pops up in how to solve the problem!

 

[Nick20]

1. Complete the square in the denominator!
2. In the numerator, write 5=2+3, and then see if an idea pops up in how to solve the problem!

Sooo if I say:

∫(x+2+3)/((x+2)^2+4)dx and substitute x+2 with u, and dx with du

∫(u+3)/(u^2+4)du am I on the right track? I'll have it done in a minute and check the answer.

thanks.

************EDIT****************

lots of stuff happened and then:

1/2 ln((x+2)^2+4) + 3/2 arctan((x+2)/2) ?

 

[Mark44]

You can check your antiderivative by differentiating it. If you end up with the original integrand, you're golden.

 

[arildno]

Sooo if I say:

∫(x+2+3)/((x+2)^2+4)dx and substitute x+2 with u, and dx with du

∫(u+3)/(u^2+4)du am I on the right track? I'll have it done in a minute and check the answer.

thanks.

************EDIT****************

lots of stuff happened and then:

1/2 ln((x+2)^2+4) + 3/2 arctan((x+2)/2) ?

The right stuff happened!

 

[R136a1]

Sooo if I say:

∫(x+2+3)/((x+2)^2+4)dx and substitute x+2 with u, and dx with du

∫(u+3)/(u^2+4)du am I on the right track? I'll have it done in a minute and check the answer.

thanks.

************EDIT****************

lots of stuff happened and then:

1/2 ln((x+2)^2+4) + 3/2 arctan((x+2)/2) ?

Don't forget the arbitrary constant +C

 

[Nick20]

Don't forget the arbitrary constant +C

Don't worry, I didn't :) I just forgot to write it here :)