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  • Need solution for RD Sharma Maths Class 12 Chapter 18 Indefiite Integrals Excercise Fill in the Blanks Question 10

Need solution for RD Sharma Maths Class 12 Chapter 18 Indefiite Integrals Excercise Fill in the Blanks Question 10

Answers (1)

Answer:

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

Hint:

To solve, this we use partial fraction.

Solution:

\begin{aligned} &I=\int \frac{x^{2}+1}{x^{2}-1} d x \\ &I=\int \frac{(x-1)^{2}+2 x}{x^{2}-1} d x \quad\quad\quad\quad\quad\quad\left[(a-b)^{2}=a^{2}+b^{2}-2 a b\right] \end{aligned}

\begin{aligned} &I=\int \frac{(x-1)^{2}}{x^{2}-1} d x+\int \frac{2 x}{x^{2}-1} d x \\ &I=\int \frac{x-1}{x+1} d x+\int \frac{2 x}{x^{2}-1} d x \end{aligned}

 

\begin{aligned} &I=\int \frac{x+1-2}{x+1} d x+\int \frac{2 x}{x^{2}-1} d x \\ &I=\int d x-2 \int \frac{1}{x+1} d x+\int \frac{2 x}{x^{2}-1} d x \\\ &I=x-2 \log |x+1|+\log \left|x^{2}-1\right|+c \end{aligned}

\begin{aligned} &I=x+\log \left|\frac{x^{2}-1}{(x+1)^{2}}\right|+c \\ &I=x+\log \left|\frac{x-1}{x+1}\right|+c \end{aligned}

 

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