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

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

Answers (1)

Answer:

            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}           

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