<s>[INST] integrate $ \frac {(x^3 + 36)} {(x^2 + 36)}$ I know I have to use long division first, but I don't really know how to do it in this case
$$\int \frac{x^3 + 36}{x^2 + 36}dx$$
 [/INST] HINT:
Using Partial fraction decomposition $$\frac{x^3+36}{x^2+36}=x+A+B\frac{\frac{d(x^2+36)}{dx}}{x^2+36}+C\frac1{x^2+36}$$
$$\implies \frac{x^3+36}{x^2+36}=x+A+\frac{2Bx}{x^2+36}+\frac C{x^2+36}$$
$$\implies x^3+36=(x+A)(x^2+36)+2Bx+C$$
$$\implies x^3+36=x^3+Ax^2+x(36+2B)+36A+C$$
Compare the coefficients of the different power of $x$  to find $A,B,C$
Now, $$\int\frac{d f(x)}{dx}dx=\int d f(x)=f(x)+C$$ and we know about  $$\int\frac{dx}{x^2+6^2}$$
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