How to calculate \int\frac{x^2}{(x^2+1)^2}dx

Julia White

Julia White

Answered question

2022-01-03

How to calculate
x2(x2+1)2dx

Answer & Explanation

Terry Ray

Terry Ray

Beginner2022-01-04Added 50 answers

First, you used parts: with u=x and dv=x(x2+1)2dx, you got
x2(x2+1)2dx=xvvdx (1)
From here, I suggested the substitution t=x2 in order to integrate dv and get v. Doing this, we see that dt=2xdx, so xdx=12dt. Thus
v=dv=x(x2+1)2dx=12dt(t+1)2
This latter integral is easy: if you can't guess it, let s=t+1. In any case, we get
v=12(t+1)=12(x2+1)
Going back to (1),
x2(x2+1)2dx=x2(x2+1)+12dxx2+1
and you should recognize the final integral as arctanx. Putting it all together, the answer is
12arctanxx2(x2+1)+C
Daniel Cormack

Daniel Cormack

Beginner2022-01-05Added 34 answers

HINT:
Use trigonometric substitution x=tanθ
x2(x2+1)2dx=12(1cos2θ)dθ
sin2θ=2tanθ1+tan2θ
Vasquez

Vasquez

Expert2022-01-09Added 669 answers

x2(x2+1)2dx=xx(x2+1)2dx[dxdxx(x2+1)2dx]dx=x11+x2+dx1+x2=...

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