Use logarithms after matching or division

With $f(x)=2x+7$ and $f^{\prime}(x)=2$, $\int\frac1{2x+7}\,dx=\frac12\int\frac2{2x+7}\,dx=\frac12\ln|2x+7|+C$, for $x\ne-7/2$.

With $f(x)=x^3+6x$ and $f^{\prime}(x)=3x^2+6=3(x^2+2)$, $\int\frac{x^2+2}{x^3+6x}\,dx=\frac13\int\frac{3(x^2+2)}{x^3+6x}\,dx=\frac13\ln|x^3+6x|+C$, for $x\ne0$.

The Teacher answer to the improper-rational-function question is long division. Divide $x^2$ by $x+1$: subtract $x(x+1)$ to leave $-x$; subtract $-(x+1)$ to leave remainder $1$. Hence $\frac{x^2}{x+1}=x-1+\frac1{x+1}$ and $\int\frac{x^2}{x+1}\,dx=\frac12x^2-x+\ln|x+1|+C$, for $x\ne-1$.