#### Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.30 Question 33

$13 \log |x|+\frac{13}{x}-12 \log |2 x+1|+C$

Hint:

To solve this integration, we use partial fraction method

Given:

$\int \frac{2 x^{2}+7 x-3}{x^{2}(2 x+1)} d x \\$

Explanation:

Let

\begin{aligned} &I=\int \frac{2 x^{2}+7 x-3}{x^{2}(2 x+1)} d x \\ &\frac{2 x^{2}+7 x-3}{x^{2}(2 x+1)}=\frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{2 x+1} \\ &2 x^{2}+7 x-3=A x(2 x+1)+B(2 x+1)+C x^{2} \\ &2 x^{2}+7 x-3=x^{2}(2 A+C)+x(A+2 B)+B \end{aligned}

Equating similar terms

$2A+C=2$                  (1)

$A+2B=7$                  (2)

$B=-3$                          (3)

Equation (2)

\begin{aligned} &A-6=7 \\ &A=13 \end{aligned}

Equation (1)

\begin{aligned} &26+C=2 \\ &C=-24 \\ &\frac{2 x^{2}+7 x-3}{x^{2}(x+1)}=\frac{13}{x}-\frac{3}{x^{2}}-\frac{24}{2 x+1} \end{aligned}

\begin{aligned} &I=13 \int \frac{d x}{x}-13 \int \frac{d x}{x^{2}}-24 \int \frac{d x}{2 x+1} \\ &I=13 \log |x|+\frac{13}{x}-24 \cdot\left(\frac{1}{2}\right) \log |2 x+1|+C \\ &I=13 \log |x|+\frac{13}{x}-12 \log |2 x+1|+C \end{aligned}