Question

If $\int \frac{x+2}{2x^2+6x+5}dx$ $=P\int \frac{4x+6}{2x^2+6x+5}dx+\frac{1}{2}\int \frac{dx}{2x^2+6x+5},$ then the values of $P$ is

• A. $\frac{1}{3}$
• B. $\frac{1}{2}$
• C. $\frac{1}{4}$
• D. $2$
• E. $1$

Answer 1

Answer: (C)

Let $x+2=A\frac{d}{dx}(2x^2+6x+5)+B$
$\Rightarrow$ $x+2=A(4x+6)+B$
$\Rightarrow$ $x+2=4Ax+6A+B$
$\Rightarrow$ $4A=1$
$\Rightarrow$ $A=\frac{1}{4}$ $6A+B=2$
$\Rightarrow$ $B=\frac{1}{2}$
$\therefore$ $\int \frac{x+2}{2x^2+6x+5}dx$ $\int \frac{\left(\frac{1}{4}(4x+6)+\frac{1}{2}\right)}{2x^2+6x+5}dx$
$=\frac{1}{4}\int \frac{4x+6}{2x^2+6x+5}dx+\frac{1}{2}\int \frac{2}{2x^2+6x+5}dx$ Comparing it with $\int \frac{x+2}{2x^2+6x+5}dx$
$=p\int \frac{4x+6}{2x^2+6x+5}dx+\frac{1}{2}\int \frac{dx}{2x^2+6x+15}$
$\Rightarrow$ $p=\frac{1}{4}$
Alternate $p(4x+6)+\frac{1}{2}=x+2$
$\Rightarrow$ $p(4x+6)=x+\frac{3}{2}$
$\Rightarrow$ $p=\frac{x+\frac{3}{2}}{4x+6}=\frac{1}{4}$