Integrate [(x^4+x^2+1)/2(1+x^2)]dx: Soln & Explanation

[Tanishq Nandan]

Homework Statement

Integrate: [(x^4+x^2+1)/2(1+x^2)]dx

Homework Equations

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Integral of x^n=x^(n+1)/(n+1)
Integral of 1/(1+x^2)=arctanx

The Attempt at a Solution

I have attached my solution.All the steps seem to be correct,but the answer isn't matching,don't know why.

 

[Fightfish]

Your integration from lines 3 to 4 is not correct:
$$\int \frac{1}{1 + x^{-2}} \mathrm{d} x \neq \tan^{-1} \left(\frac{1}{x}\right).$$
My suggestion is to instead split the original expression as
$$\frac{x^4 + x^2 + 1}{2(1 + x^{2})} = \frac{x^4 + x^2}{2(1 + x^{2})} + \frac{1}{2(1 + x^{2})}.$$

 

[Tanishq Nandan]

My suggestion is to instead split the original expression as

x4+x2+12(1+x2)=x4+x22(1+x2)+12(1+x2).x4+x2+12(1+x2)=x4+x22(1+x2)+12(1+x2).​

Oops,why didn't I think of that?Thanks.

∫11+x−2dx≠tan−1(1x).∫11+x−2dx≠tan−1⁡(1x).​

\int \frac{1}{1 + x^{-2}} \mathrm{d} x \neq \tan^{-1} \left(\frac{1}{x}\right).

Yep,sorry.