Integration Probs: int(6x+10)/(x^2+4)dx, int 1/(16+x^2)^2dx,

[Eternal Sky]

I am attempting to evaluate the following integrals, but I'm not quite sure how to proceed. The integrals are:

(6x + 10)/(x^2 + 4) dx

dx/(16 + x^2)^2

dx/x(x^2 + 1)

I'm pretty sure that I need to use trig substitutions on the first two, but I'm not exactly sure how.

Any help is greatly appreciated.

 

[soroban]

Re: Integration Problems

Hello, Eternal Sky!

\(\displaystyle \int\frac{6x + 10}{x^2 + 4}\,dx\)

\(\displaystyle \text{Make two integrals: }\;\int\frac{6x}{x^2+4}\,dx + \int\frac{10}{x^2+4}\,dx\)

\(\displaystyle \text{The first is }\ln\text{, the second is }\arctan\)

\(\displaystyle \int \frac{dx}{(16 + x^2)^2}\)

\(\displaystyle \text{Let }\,x \:=\:4\tan\theta\quad\Rightarrow\quad dx \:=\:4\sec^2\!\theta\,d\theta\quad\hdots\;\;\text{and substitute.}\)

\(\displaystyle \int\frac{dx}{x(x^2 + 1)}\)

I must assume you are familiar with Partial Fractions.

. . \(\displaystyle \frac{1}{x(x^2+1)} \;=\;\frac{A}{x} + \frac{Bx+C}{x^2+1}\)

\(\displaystyle \text{You should get: }\:A = 1,\;B = -1,\;C = 0\)

 

[Eternal Sky]

Re: Integration Problems

I finally understand now.

Thank you so much for your help!