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Provide Solution For  R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise 18.20 Question 11 Maths Textbook Solution.

Answers (1)

Answer:x+\log \left|\frac{x}{x+1}\right|+c

Given: \int \frac{x^{3}+1}{x^{3}-x} d x

Hint: using Partial Fraction and \int \frac{1}{x} d x

Explanation: Let

                     I=\int \frac{x^{3}+1}{x^{3}-x} d x



\therefore \int \frac{x^{3}+1}{x^{3}-x} d x=\int 1 d x+\int \frac{1}{x(x+1)} d x

                              =x+I_{1}   ...................(1) Where I_{1}=\int \frac{1}{x\left ( x+1 \right )}dx


Multiplying by x\left ( x+1 \right )

1=A\left ( x+1 \right )+B\left ( x+1 \right )

Putting x = -1

1=A\left ( 0+1 \right )+B\left ( 0 \right )\Rightarrow A=1

\begin{aligned} &\frac{1}{x(x+1)}=\frac{1}{x}-\frac{1}{(x+1)} \\ &\therefore \int \frac{1}{x(x+1)} d x=\int \frac{1}{x} d x-\int \frac{1}{x+1} d x \end{aligned}

                                   =\log |x|-\log |x+1|

Put in (1)

\begin{aligned} I &=x+\log |x|-\log |x+1|+c \\ &=x+\log \left|\frac{x}{x+1}\right|+c \end{aligned}

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