Hints on how to do the following integrals...

[Guest]

The integral for (x^2)/(1-x^4)

The integral of Square root of (x+1)/Square root of (x-1)

The integral of (3x+2)/((x(x+1)^3)

thank you in advance...

 

[galactus]

For the first one, use partial fractions.

\(\displaystyle \L\\\frac{A}{x+1}+\frac{B}{x-1}+\frac{C}{x^{2}+1}=x^{2}\)

The second one is trickier.

Make the substitution, \(\displaystyle u=\sqrt{x-1};\;\ 2du=\frac{1}{\sqrt{x-1}}dx;\;\ x=u^{2}+1\)

You get:

\(\displaystyle \L\\2\int\sqrt{2+u^{2}}du\)

Now, you could use trig sub:

Let \(\displaystyle u=\sqrt{2}tan({\theta});\;\ du=\sqrt{2}sec^{2}(\theta)d{\theta}\)




The third one, partial fractions again.

\(\displaystyle \L\\\frac{A}{x}+\frac{B}{x+1}+\frac{C}{(x+1)^{2}}+\frac{D}{(x+1)^{3}}=3x+2\)