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  • explain solution RD Sharma class 12 Chapter 18 Indefinite Integrals exercise 18.30 question 19

explain solution RD Sharma class 12 Chapter 18 Indefinite Integrals exercise 18.30 question 19

Answers (1)

Answer:

            \log \left|\frac{\left(x^{2}-1\right)^{3}}{x}\right|+C

Hint:

            To solve this integration, we use partial fraction method   

Given:

            \int \frac{5 x^{2}-1}{x(x-1)(x+1)} d x

Explanation:

Let

I=\int \frac{5 x^{2}-1}{x(x-1)(x+1)} d x
\begin{aligned} &\frac{5 x^{2}+1}{x(x-1)(x+1)}=\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x+1} \\ &5 x^{2}+1=A(x-1)(x+1)+B(x)(x+1)+C(x)(x-1) \\ &\text { At } x=1 \\ &5(1)+1=A(0)+B(1)(2)+C(0) \\ &6=2 B \\ &B=3 \end{aligned}
\begin{aligned} &\text { At } x=-1 \\ &5(1)+1=A(0)+B(0)+C(-1)(-2) \\ &6=2 C \\ &C=3 \end{aligned}
\begin{aligned} &\text { At } x=0 \\ &5(0)+1=A(-1)(1)+0+0 \\ &1=-A \\ &A=-1 \end{aligned}
\begin{aligned} &\frac{5 x^{2}-1}{x(x-1)(x+1)}=\frac{-1}{x}+\frac{3}{x-1}+\frac{3}{x+1} \\ &I=-\int \frac{1}{x} d x+3 \int \frac{1}{x-1} d x+3 \int \frac{1}{x+1} d x \\ &=-\log |x|+3 \log |x-1|+3 \log |x+1|+C \end{aligned}

Using the formulas,(x+y)(x-y)=x^{2}-y^{2} \& \log x+\log y=\log\left ( xy \right )
I=\log \left|\frac{\left(x^{2}-1\right)^{3}}{x}\right|+C

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