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  • Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise Multiple Choice Questions Question 34 Maths Textbook Solution.

Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise Multiple Choice Questions  Question 34 Maths Textbook Solution.

Answers (1)

Answer:

x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\log |1+x|+C

Given:

\int \frac{x^{3}}{x+1} d x

Hint:

You must know how to integrate \int x^{n} d xand \int \frac{1}{x} d x

Explanation:

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

         \begin{aligned} &=\int \frac{x^{3}+1-1}{x+1} d x \\ &=\int \frac{(x+1)\left(x^{2}+1-x\right)}{x+1} d x-\int \frac{1}{x+1} d x \end{aligned}                                                    \left[\because x^{3}+1=(x+1)\left(x^{2}+1-x\right)\right]

        \begin{aligned} &=\int\left(x^{2}+1-x\right) d x-\int \frac{1}{1+x} d x \\ &=\frac{x^{3}}{3}+x-\frac{x^{2}}{2}-\log |1+x|+C \\ &=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\log |1+x|+C \end{aligned}

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