1.24 Example continued

Thus we have to integrate $\frac{1}{x-1}$ and $\frac{2x+2}{x^{2}+9}$. The integral of the first term is just $\log|x-1|+c$.

We further split the second term into $\frac{2x}{x^{2}+9}$ and $\frac{2}{x^{2}+9}$. For the first of these, we make the substitution $u={x^{2}+9}$ to obtain that the integral is $\log(x^{2}+9)+c^{\prime}$.

For the second, we substitute $x={3\tan t}$ to obtain the integral

Putting all of these together, we obtain: