# Integrate (x^3 + x^2)/(1 + x^4) using substitution?

• ritwik06

#### ritwik06

Integrate the following:

(x^3+x^2)/(1+x^4)

I have been taught only integration by substitution. My teacher told me that this can be solved using that ith some trick.

I have tried for a long time. All that I can do was to convert the numerator to x^2(x+1)
and the denominator to (x^2-1)^2+2x^2
but without sucess.

Next I tried to break the given thing into two terms. it helped a little but I was again stuck on the second term, ie. , i couldn't again find out the integration of x^2/(1+x^4)

Is it easy to find the integral of (x^2)/(1+x^4)

I have not tried this, but you could try factoring 1+x^4 into two quadratics, then do partial fraction decomposition.

$$\int\frac{x^{2}}{1+x^{4}}$$

Can this be determined?

I have not tried this, but you could try factoring 1+x^4 into two quadratics, then do partial fraction decomposition.

its difficult to factorize $$x^{4}+1$$.
Moreover,
can u please tell me about partial fractions decomposition? I haven't learned it yet.

$$\int\frac{x^{2}}{1+x^{4}}$$

Can this be determined?

Have you tried the obvious substitution u= x4+ 1??

Have you tried the obvious substitution u= x4+ 1??
yes,
it yields the following:
0.25$$\int\frac{1}{u \sqrt[4]{(u-1)}}$$du
But does this help or more complicate the problem?

Last edited:

$$1+x^4 = 1+2x^2+x^4 - 2x^2=(1+x^2)^2-2x^2=(1-\sqrt 2 x + x^2)(1+\sqrt 2 x+x^2)$$.

As to partial fraction decomposition, I suggest you have a look in your textbook or at wikipedia or other webpages.

$$1+x^4 = 1+2x^2+x^4 - 2x^2=(1+x^2)^2-2x^2=(1-\sqrt 2 x + x^2)(1+\sqrt 2 x+x^2)$$.

As to partial fraction decomposition, I suggest you have a look in your textbook or at wikipedia or other webpages.

Yeah, I have read now about partial fractions. It means to cahnge the given thing into sum of 2 fractions. But hile trying to change:
$$\frac{x^{2}}{1+x^{4}}$$
in the form of
$$\frac{A}{1-x\sqrt{2}+x^{2}}$$+$$\frac{B}{1+x\sqrt{2}+x^{2}}$$
I get the following equation:
$$x^{2}$$=(A+B)x$$^{2}$$+(A+B)(2$$\sqrt{x}$$+1)

I cannot solve further for A and B. Can u do it?

Its one hell out of the other. What should I do? Please help me! and try it out before answering. Thanks for all efforts!

Anyone going to help me out? Please!

Your fractions should be of the form:
$$\frac{Ax+B}{1-x\sqrt{2}+x^{2}} + \frac{Cx+D}{1+x\sqrt{2}+x^{2}}$$.