<s>[INST] $\int\frac1{(x^2+1)^2}\ dx$ by partial fraction decomposition Is there any possible way to calculate the integral of $\frac{1}{(x^2+1)^2}$ by partial fraction decomposition? I do not know the formulas for the trigonometric method.
Thank you!
 [/INST] When you let $x=\tan\theta$, and $dx=\dfrac{d\theta}{\cos^2\theta}$
$$\int\dfrac{1}{(1+x^2)^2}dx=\int\dfrac{1}{(1+\tan^2\theta)^2\cos^2\theta}d\theta=\int\dfrac{\cos^4\theta}{\cos^2\theta}d\theta=\int\cos^2\theta d\theta\\=\dfrac{\theta}{2}+\dfrac{\sin^2\theta}{4}+C=\dfrac{\arctan x}{2}+\dfrac{x}{2(1+x^2)}+C$$
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