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  • Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18 .14 Question 9

Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.14 Question 9

Answers (1)

Answer:  -\log \left|(2-x)+\sqrt{(2-x)^{2}-1}\right|+c

Hint: To solve this integral, use special integral formula.

Given:   \int \frac{1}{\sqrt{(2-x)^{2}-1}} d x \\

Solution:

Let

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

\text { Put } 2-x=t \Rightarrow-d x=d t \Rightarrow d x=-d t \text { then }

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

=-\log \left|t+\sqrt{t^{2}-1}\right|+c \quad\quad\quad\quad\quad\quad\left[\because \int \frac{1}{\sqrt{x^{2}-a^{2}}} d x=\log \left|x+\sqrt{x^{2}-a^{2}}\right|+c\right]

=-\log \left|(2-x)+\sqrt{(2-x)^{2}-1}\right|+c \quad \quad \quad \quad \quad[\because t=2-x]

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