I'm struggling w/ this partial fraction. I have an exam tomorrow, I've lost hope. Reddit r/calculus

How to integrate (x^2+5x+1)/(x^2+1)^2 by using partial fraction decomposition? This is a must-learn technique for your Calculus 2 class. Subscribe to @bprpcalculusbasics for more calculus tutorials.

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[goldenwarrior]

I actually did it with trig sub(x=tanΘ) and I got my answer as arctan(x) - (5/4)({1-x^2}/{1+x^2}) + C. Y'all should try this approach too!

[dentonyoung]

That brings back memories of doing that in calculus class, many years ago. I still love watching these, it's good practice for my brain.

[paulnokleberg]

I just noticed something strange when integrating by trig sub, with x=tan(theta). I get arctanx+5(integral of Using u=sin(theta), I get arctanx + (5/2)((x^2)/(1+(x^2))+C, but using u=cos(theta), I get arctanx - (5/2)(1/(1+(x^2))+C, the same as the original method. Both trig sub methods with different u-subs are valid methods, but give different answers that, interestingly, have the same derivative, equal to the original integrand.
Could you or anyone elaborate further on this? It seems strange to me.

[沈博智-xy]

Looking at their work, they just made a careless mistake when using B = 1 and B + D = 1 => 1 + D = 1 => D = 2 [sic] (should be D = 0)

otherwise their partial fractions is all correct.