Question

How do you integrate `1/((1+x^2)^2)`?

Answer 1

Reqd. Integral `=1/2{arctanx+x/(1+x^2)}.`

Explanation:

Take the substitution `x=tant`
`:. dx=sec^2tdt.`
`:. int{1/((1+x^2)^2)}dx =int{1/(1+tan^2t)^2}sec^2tdt=int(1/sec^4t)sec^2tdt=int(cos^4t/cos^2t)dt=intcos^2tdt=int(1+cos2t)/2dt=1/2{t+(sin2t)/2)`

Now `x=tant rArr t=arc tanx.` Further, `sin2t =(2tant)/(1+tan^2t)=(2x)/(1+x^2).`

`:.` Reqd. Integral `=1/2{arctanx+x/(1+x^2)}.`