Question

$\int \frac{3x+5}{x^2+3x-18}dx$

Answer 1

$I=\int \frac{3x+5}{x^2+3x-18}dx$

$\Rightarrow I=3\int \frac{x+{\displaystyle \frac{5}{3}}}{x^2+3x-18}dx$

$\Rightarrow I=\frac{3}{2}\int \frac{2x+{\displaystyle \frac{10}{3}}}{x^2+3x-18}dx$

$\Rightarrow I=\frac{3}{2}\int \frac{2x+3+{\displaystyle \frac{1}{3}}}{x^2+3x-18}dx$

$\Rightarrow I=\frac{3}{2}\int \frac{2x+3}{x^2+3x-18}dx+\frac{1}{2}\int \frac{1}{x^2+3x-18}dx$

$\Rightarrow I=\frac{3}{2}\log \left|x^2+3x-18\right|+\frac{1}{2}\int \frac{1}{x^2+3x+{\displaystyle \frac{9}{4}}-18-{\displaystyle \frac{9}{4}}}dx$ $\left(\begin{array}{c}\int \frac{f\text{'}\left(x\right)dx}{f\left(x\right)}=\log \left|f\left(x\right)\right|+C\\ f\left(x\right)=x^2+3x-18\Rightarrow f\text{'}\left(x\right)=2x=3\end{array}\right)$

$\Rightarrow I=\frac{3}{2}\log \left|x^2+3x-18\right|+\frac{1}{2}\int \frac{1}{{\left(x+{\displaystyle \frac{3}{2}}\right)}^2-{\displaystyle \frac{81}{4}}}dx$

$\Rightarrow I=\frac{3}{2}\log \left|x^2+3x-18\right|+\frac{1}{2}\int \frac{1}{{\left(x+{\displaystyle \frac{3}{2}}\right)}^2-{\displaystyle {\left(\frac{9}{2}\right)}^2}}dx$

$\Rightarrow I=\frac{3}{2}\log \left|x^2+3x-18\right|+\frac{1}{2}\times \frac{1}{2\times {\displaystyle \frac{9}{2}}}\log \left|\frac{\left(x+{\displaystyle \frac{3}{2}}\right)-{\displaystyle \frac{9}{2}}}{\left(x+{\displaystyle \frac{3}{2}}\right)+{\displaystyle \frac{9}{2}}}\right|+C$ $\left(\int \frac{dx}{x^2-a^2}=\frac{1}{2a}\log \left|\frac{x-a}{x+a}\right|+C\right)$

$\Rightarrow I=\frac{3}{2}\log \left|x^2+3x-18\right|+\frac{1}{18}\log \left|\frac{x-3}{x+6}\right|+C$