Thread: Working with Different Integration Techniques

1. Working with Different Integration Techniques

Alright, so I've got a homework assignment that I've been working through and I've come across several problems that are tripping me up. If anyone could offer suggestions as to where I should start with these, it'd be greatly appreciated.

#1) Evaluate the following integral:

[xe^arctan(x) / (1+x^2)^(3/2)] dx

#2) Evaluate the following integrals of rational functions:

a) Integral of [1 / (4x^2+4x+1)] dx
b) Integral of [1 / (4x^2+4x+5)] dx

#3) I'm supposed to use the substitution t = tan(x/2) to convert the following integral into a rational function integral. (Put it in the lowest terms and then evaluate):

Integral of:

[1 / 1+sin(x)-cos(x)] dx

Alright, so as far as what I've tried thus far:

1) I didn't even know where to start...
2) I figured that partial fractions would be the way to go here, so I broke up the integrals as per standard procedure and began to integrate, but when I finished that train of thought, I ended up getting something way different than the few people I asked (too long to even bother posting here)...
3) Here, I was supposed to use a similar substitution as was used in a previous problem, but when I did, I ended up getting another weird answer, so I figured maybe some fresh eyes without any background info might be helpful to see something that I was missing...

Thanks again.

-B

P.S. - I'm not digging for free help here, I just wanted some pointers on what I can do to start these and actually end up with the correct answer...

2. I can help with the first one:

Use the substitution: $\displaystyle x = \tan u$ or $\displaystyle u = \arctan x$

Then,

$\displaystyle du = \frac{1}{1+x^2}dx$ but also $\displaystyle dx = \sec ^2 (u)du$

Apparently,

$\displaystyle (1+x^2) = \sec ^2 u$

Then we have:

$\displaystyle \frac{e^{\tan ^{-1}(x)} x}{\left(x^2+1\right)^{3/2}}dx = \frac{\tan (u) e^{u}}{\sec u}du = \sin (u) e^u du$

If you know the integral of $\displaystyle \sin (u) e^u du$ then you are good to go. Otherwise you can split up sine as:

$\displaystyle \sin u = \frac{e^{iu}-e^{-iu}}{2i}$ and proceed from there.

3. Alright, I think I see where you're going with this. Thanks for the help...

Any ideas as to where I should begin with the others?

-B

4. For #2, notice that the denominator in a) is just (1+2x)^2, so this one turns out to be really simple.