integration questions (1 Viewer)

[CriminalCrab]

please help solve
integrate:
by substitution:
1) dx/(x sqrt(x^2-4)) [x=1/u]
2) dx/[(a^2-x^2)^(3/2)] [x=asintheta]
3)(e^2x)dx/(e^4x -1) [u=e^2x]
4) cosx dx /(1+sinx)(2+sinx)

by parts:
1)cos^4(x)
2)x(inverse tanx)
3) log(a^2+x^2)

and....write as partial fraction:
1/(x^4 +1)

Thanks in advance (they are playing guitar and cooking for you)

 

[Carrotsticks]

What is the problem with the substitution ones? The hardest part (if you can't see it) is finding the appropriate substitution but after that it's just simple algebra.

For the Parts ones:

1. Don't need IBP, you can split it to become (1-sin^2(x))cos^2(x), then expand, then integrate each part separately (need to use Double Angle formulae).

2. u = arctan(x) and dv = x

3. u = log(a^2+x^2) and dv = x.

And for the last one... that expression is irreducible. Did you want us to decompose it to complex linear factors?

 

[nightweaver066]

For that last one, x^4 + 1 = x^4 + 2x^2 + 1 - 2x^2 = (x^2 + 1)^2 - 2x^2 = (x^2 + sqrt(2)x + 1)(x^2 - sqrt(2)x - 1)

Partial fractions from there.

 

[Carrotsticks]

Correction: Irreducible over haha!

 

[juantheron]