<s>[INST] Partial fraction integral Question:
$\int \dfrac{5 }{(x+1) (x^2 + 4)  } dx $ 
Thought process: 
I'm treating it as a partial fraction since it certainly looks like one.
I cannot seem to solve it besides looking at it in the "partial fraction" way.
My work:
1) Focus on the fraction part first ignoring the $\int $ and $dx$ for the moment.
Multiply $(x+1) (x^2+4)$ on both sides of the equation and get: 
$5$ =  $\dfrac{A }{(x+1) } $ +  $\dfrac{Bx+C }{(x^2+4) } $ 
Note: the x^2 + 4 is irreducible which explains the Bx+c as the numerator.
$ 5 = A(x^2+ 4) + Bx + C(x+1) $
I tried x = -1 which knocks out C:
$5 = A5 + - B $
I also tried x = 0 which knocks out B
$5 = A4 + C$ 
A is a lot harder to knock out since the squared changes the picked value to be positive.
I decided to add the two found equation together and get
$10 = A9 - B + C $
Now I am officially stuck at this step.
 [/INST] Split the fraction. 
$$\color{blue}{\dfrac{5 }{(x+1)(x^2 + 4)} = \dfrac{P(x)}{(x+1)}+\dfrac{Q(x)}{(x^2+4)}}$$
The largest partial fraction's denominator is an order 2 polynomial, so the order of the partial fractions' numerators must become order 2 when cross multiplied with the other partial fractions' denominators.  Ie: P(X) is order 0, Q(X) is order 1.
$$\color{blue}{\dfrac{5 }{(x+1)(x^2 + 4)} = \dfrac{A}{(x+1)}+\dfrac{Bx+C}{(x^2+4)}}$$
Now that you know what the target looks like, perform the crossmultiplication.
$$\color{blue}{5 = A(x^2+4)+(Bx+C)(x+1)}$$
Expand the polynomial so we can compare the coefficients.
$$\color{blue}{5 = (A+B)x^2 + (B+C)x + (4A+C)}$$
Which allows us to conclude
$$\color{blue}{A=-B, C=-B, 5= 4A+C \implies B=-1, A=1, C=1}$$
So we have our partial fraction expansion.
$$\color{blue}{\therefore \dfrac{5 }{(x+1)(x^2 + 4)} = \dfrac{1}{(x+1)}+\dfrac{(x+1)}{(x^2+4)}}$$
Next use: $\color{blue}{\dfrac{\mathrm{d}}{\mathrm{d}x}\ln(x+a) = \dfrac{1}{x+a}}$, $\color{blue}{\dfrac{\mathrm{d}}{\mathrm{d}x}\ln(x^2+b^2)= 2\dfrac{x}{x^2+b^2}}$ and $\color{blue}{\dfrac{\mathrm{d}}{\mathrm{d}x} \arctan \dfrac{x}{c} = \dfrac{c}{x^2+c^2}}$
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