Question

Integrate with respect to x , (3x+7)/(2x²+3x-2)

Answer 1

First we've to factorise the denominator.

$\displaystyle\longrightarrow\sf{\int\dfrac{3x+7}{2x^2+3x-2}\ dx=\int\dfrac{3x+7}{2x^2+4x-x-2}\ dx}$

$\displaystyle\longrightarrow\sf{\int\dfrac{3x+7}{2x^2+3x-2}\ dx=\int\dfrac{3x+7}{2x(x+2)-(x+2)}\ dx}$

$\displaystyle\longrightarrow\sf{\int\dfrac{3x+7}{2x^2+3x-2}\ dx=\int\dfrac{3x+7}{(x+2)(2x-1)}\ dx}$

Splitting the numerator as follows:

$\displaystyle\longrightarrow\sf{\int\dfrac{3x+7}{2x^2+3x-2}\ dx=\int\dfrac{x+2+2x-1+6}{(x+2)(2x-1)}\ dx}$

$\displaystyle\longrightarrow\sf{\int\dfrac{3x+7}{2x^2+3x-2}\ dx=\int\left(\dfrac{x+2+2x-1}{(x+2)(2x-1)}+\dfrac{6}{(x+2)(2x-1)}\right)\ dx}$

$\displaystyle\longrightarrow\sf{\int\dfrac{3x+7}{2x^2+3x-2}\ dx=\int\left(\dfrac{1}{x+2}+\dfrac{1}{2x-1}+\dfrac{6}{(x+2)(2x-1)}\right)\ dx}$

Distributing integral to each term,

$\displaystyle\begin{aligned}\longrightarrow\sf{\int\dfrac{3x+7}{2x^2+3x-2}\ dx}=\ \ &\sf{\int\dfrac{1}{x+2}\ dx+\dfrac{1}{2}\int\dfrac{2}{2x-1}\ dx}\\\\+\ \ &\sf{6\int\dfrac{1}{(x+2)(2x-1)}\ dx}\end{aligned}$

$\displaystyle\begin{aligned}\longrightarrow\sf{\int\dfrac{3x+7}{2x^2+3x-2}\ dx}=\ \ &\sf{\ln|x+2|+\dfrac{1}{2}\ln|2x-1|}\\\\+\ \ &\sf{6\int\dfrac{1}{(x+2)(2x-1)}\ dx\quad\quad\dots(1)}\end{aligned}$

Let,

$\longrightarrow\sf{\dfrac{1}{(x+2)(2x-1)}=\dfrac{A}{x+2}+\dfrac{B}{2x-1}}$

$\longrightarrow\sf{\dfrac{1}{(x+2)(2x-1)}=\dfrac{A(2x-1)+B(x+2)}{(x+2)(2x-1)}}$

$\longrightarrow\sf{A(2x-1)+B(x+2)=1}$

$\longrightarrow\sf{(2A+B)x+(2B-A)=1}$

Equating corresponding coefficients,

$\longrightarrow\sf{2A+B=0}$

$\longrightarrow\sf{2B-A=1}$

On solving them we get,

$\longrightarrow\sf{A=-\dfrac{1}{5}}$

$\longrightarrow\sf{B=\dfrac{2}{5}}$

Hence (1) becomes,

$\displaystyle\begin{aligned}\longrightarrow\sf{\int\dfrac{3x+7}{2x^2+3x-2}\ dx}=\ \ &\sf{\ln|x+2|+\dfrac{1}{2}\ln|2x-1|}\\\\+\ \ &\sf{6\int\left(-\dfrac{1}{5(x+2)}+\dfrac{2}{5(2x-1)}\right)\ dx}\end{aligned}$

$\displaystyle\begin{aligned}\longrightarrow\sf{\int\dfrac{3x+7}{2x^2+3x-2}\ dx}=\ \ &\sf{\ln|x+2|+\dfrac{1}{2}\ln|2x-1|}\\\\-\ \ &\sf{\dfrac{6}{5}\int\dfrac{1}{x+2}\ dx+\dfrac{6}{5}\int\dfrac{2}{2x-1}\ dx}\end{aligned}$

$\displaystyle\longrightarrow\sf{\int\dfrac{3x+7}{2x^2+3x-2}\ dx=\ln|x+2|+\dfrac{1}{2}\ln|2x-1|-\dfrac{6}{5}\ln|x+2|+\dfrac{6}{5}\ln|2x-1|+c}$

$\displaystyle\longrightarrow\underline{\underline{\sf{\int\dfrac{3x+7}{2x^2+3x-2}\ dx=\dfrac{17}{10}\ln|2x-1|-\dfrac{1}{5}\ln|x+2|+c}}}$