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  • explain solution RD Sharma class 12 Chapter 18 Indefinite Integrals exercise 18.30 question 47

explain solution RD Sharma class 12 Chapter 18 Indefinite Integrals exercise 18.30 question 47

Answers (1)

best_answer

Answer:

                \frac{1}{8} \log x-\frac{1}{24} \log \left(x^{3}+8\right)+C

Hint:

            To solve this integration, we use partial fraction method   

Given:

            \int \frac{1}{x\left(x^{3}+8\right)} d x \\

Explanation:

Let

\begin{aligned} &I=\int \frac{1}{x\left(x^{3}+8\right)} d x \\ &I=\int \frac{x^{2}}{x^{3}\left(x^{3}+8\right)} d x \end{aligned}                                       [Multiply and divide byx^{2}]

Let

\begin{aligned} &x^{3}=y \\ &3 x^{2} d x=d y \\ &x^{2} d x=\frac{d y}{3} \\ &I=\frac{1}{3} \int \frac{d y}{y(y+8)} \\ &\frac{1}{y(y+8)}=\frac{A}{y}+\frac{B}{y+8} \\ &1=A(y+8)+B y \\ &1=(A+B) y+8 A \end{aligned}

Equating both side

\begin{aligned} &8 A=1 \\ &A=\frac{1}{8} \\ &A+B=0 \\ &B=-A=\frac{-1}{8} \\ &\frac{1}{y(y+8)}=\frac{1}{8 y}-\frac{1}{8(y+8)} \end{aligned}

Thus

\begin{aligned} &I=\frac{1}{3} \int\left(\frac{1}{8 y}-\frac{1}{8(y+8)}\right) d y \\ &I=\frac{1}{24} \int \frac{1}{y} d y-\frac{1}{24} \int \frac{1}{y+8} d y \end{aligned}

\begin{aligned} &I=\frac{1}{24} \log |y|-\frac{1}{24} \log |y+8|+C \\ &I=\frac{1}{24} \log \left|x^{3}\right|-\frac{1}{24} \log \left|x^{3}+8\right|+C \\ &I=\frac{1}{8} \log |x|-\frac{1}{24} \log \left|x^{3}+8\right|+C \end{aligned}

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