# Help on analytical inegration

Hi all.
Can someone help me on how to analytically evaluate the following integral:

integral of f(x) with respect to x, between limits 1 and 0 where

f(x)= x^4/(1+x^2)

I have thought about an x=tan u substitution but then i am left with a (tan u)^4 integral, and i dont know how to do that. Anyway, I don’t think it is the optimum method.

TIA

u=1+x^2
du = 2xdx

after some playing with it

x^2=u-1 // helpful because (x^2)^2 = x^4

f(0)=1
f(1)=2

so for u from 1 to 2, f(u)=(u-1)^2/u

is that easier to integrate?

Hey, i can speak greek too

ckin, ur greek seems a little rusty - the simplified integral is still with respect to x

u are forgetting that
du/dx = 2x

so dx = du/2x

(to make integral with respect to u)

so the integral u are left with is

(u-1)^1.5 / u

Originally posted by shuebhussain
[B]u are forgetting that
du/dx = 2x

so dx = du/2x [/B]
well aint that the same with
Originally posted by ckin2001
du = 2xdx
??

ah yes you are right ; thanks for pointing that out. But the point i made in my last post still stands and that is that a u=1+x^2 will get you no where because (u-1)^1.5 / u isn’t easy to integrate. (Thanks for your efforts anyway)

But, i have cracked it !

Re-write f(x)= x^4/(1+x^2) as

f(x)= (x^4 - 1 + 1 )/(1+x^2)
= (x^4 - 1)/(1+x^2) + 1/(1+x^2)
= (x^2 + 1)(x^2 - 1)/(1+x^2) + 1/(1+x^2)
= (x^2 - 1) + 1/(1+x^2)

Thus the integral of f(x) w.r.t. x
=x^3/3 - x + arctanx + constant

//tried

i didnt forget hemi - but i forgot to incorporate. thats why i have mathematica on this pc.

whats mathematica

Originally posted by shuebhussain
whats mathematica