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  • Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.14 Question 8

Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.14 Question 8

Answers (1)

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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

I=\int \frac{1}{\sqrt{(2-x)^{2}+1}} d x

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

I=\int \frac{1}{\sqrt{t^{2}+1}}(-d t)=-\int \frac{1}{\sqrt{t^{2}+1^{2}}} d t

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

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

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