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  • explain solution RD Sharma class 12 Chapter 18 Indefinite Integrals exercise 18.30 question 26

explain solution RD Sharma class 12 Chapter 18 Indefinite Integrals exercise 18.30 question 26

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Answer:

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

Hint:

            To solve this integration, we use partial fraction method   

Given:

            \int \frac{x^{3}+x+1}{x^{2}-1} d x

Explanation:

Let

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

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