How do you integrate $int ( 1/((x+1)^2+4))$ using partial fractions?

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Answer 1 · 2016-09-12T18:55:52

$1/2arctan((x+1)/2)+C$

This cannot be expressed using partial fractions. However, we can integrate this using trigonometric substitutions.

$intdx/((x+1)^2+4)$

Let $x+1=2tantheta$ . Thus, $dx=2sec^2thetad theta$ . Substituting, this gives us:

$=int(2sec^2thetad theta)/(4tan^2theta+4)$

Factoring:

$=1/2int(sec^2thetad theta)/(tan^2theta+1)$

Recall that $tan^2theta+1=sec^2theta$ :

$=1/2int(sec^2thetad theta)/sec^2theta$

$=1/2intd theta$

$=1/2theta+C$

From $x+1=2tantheta$ we see that $theta=arctan((x+1)/2)$ :

$=1/2arctan((x+1)/2)+C$

How do you integrate int ( 1/((x+1)^2+4)) int ( 1/((x+1)^2+4)) using partial fractions?