<s>[INST] Integration Question on $\int \frac{1}{x^2+10x+21} \, dx$ How would i do the following indefinite integration $$\int \frac{1}{x^2+10x+21} \, dx$$
so far I've turned the bottom polynomial into $(x+7)(x+3)$ not too sure where to go from here
 [/INST] Hint: Partial Fractions:
$$\frac{1}{(x+7)(x+3)}=\frac{A}{x+7}+\frac{B}{x+3}$$
Now find $A,B$ and use the linearity of the integral:
$$\int \frac{1}{(x+7)(x+3)} dx=\int \frac{A}{x+7}dx+\int \frac{B}{x+3}dx$$
This should be simple now.
EDIT: Evaluating $A,B$:
$$\frac{1}{(x+7)(x+3)}=\frac{A}{x+7}+\frac{B}{x+3}\iff 1=A(x+3)+B(x+7)$$
Setting $x=-3$ and $x=-7$ give $A,B=?$
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