# Get Answers to all your Questions

### Answers (1)

Answer: $\log \left|x^{2}-x-2\right|+2 \log \left|\frac{x-2}{x+1}\right|+c$

Hint: You must know about how to solve integration

Given:$\int \frac{2 x+5}{x^{2}-x-2} d x$

Solution:

Let $I=\int \frac{2 x+5}{x^{2}-x-2} d x$

$=\int \frac{2 x-1+6}{x^{2}-x-2} d x$

$=\int \frac{2 x-1}{x^{2}-x-2} d x+\int \frac{6}{x^{2}-x-2} d x$

$I_{1}=\int \frac{2 x-1}{x^{2}-x-2} d x$

Let

$x^{2}-x-2=t$

$\Rightarrow(2 x-1) d x=d t$

\begin{aligned} &I_{1}=\int \frac{d t}{t}=\log |t|+c_{1} \\ &=\log \left|x^{2}-x-2\right|+c_{1} \end{aligned}

\begin{aligned} &I_{2}=\int \frac{6}{x^{2}-x-2} d x \\ &=6 \int \frac{d x}{x^{2}-x+\frac{1}{4}-\frac{1}{4}-2} \\ &=6 \int \frac{d x}{\left(x-\frac{1}{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}}\left[\int \frac{d x}{x^{2}-a^{2}}=\frac{1}{2 a} \log \left|\frac{x-a}{x+a}\right|+c\right] \\ &=6 \times \frac{1}{3} \log \left|\frac{x-\frac{1}{2}-\frac{3}{2}}{x-\frac{1}{2}+\frac{3}{2}}\right|+c_{2} \\ &=2 \log \left|\frac{2 x-4}{2 x+2}\right|+c_{2} \end{aligned}

So,$I=I_{1}+I_{2}$

$=\log \left|x^{2}-x-2\right|+2 \log \left|\frac{x-2}{x+1}\right|+c$

View full answer

## Crack CUET with india's "Best Teachers"

• HD Video Lectures
• Unlimited Mock Tests
• Faculty Support