Question

compute $int\frac{x + 5}{x^{2}+9}dx$.

Explanation:

Step1: Split the integral

Split $\int\frac{x + 5}{x^{2}+9}dx$ into $\int\frac{x}{x^{2}+9}dx+\int\frac{5}{x^{2}+9}dx$.

Step2: Solve $\int\frac{x}{x^{2}+9}dx$

Let $u=x^{2}+9$, then $du = 2x dx$, and $\int\frac{x}{x^{2}+9}dx=\frac{1}{2}\int\frac{du}{u}=\frac{1}{2}\ln|u|=\frac{1}{2}\ln|x^{2}+9|$.

Step3: Solve $\int\frac{5}{x^{2}+9}dx$

Since $\int\frac{1}{a^{2}+x^{2}}dx=\frac{1}{a}\arctan(\frac{x}{a})+C$ (here $a = 3$), then $\int\frac{5}{x^{2}+9}dx=5\times\frac{1}{3}\arctan(\frac{x}{3})=\frac{5}{3}\arctan(\frac{x}{3})$.

Answer:

$\frac{1}{2}\ln|x^{2}+9|+\frac{5}{3}\arctan(\frac{x}{3})+C$