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Provide Solution For R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise  Revision Exercise Question 42 Maths Textbook Solution.

Answers (1)

Answer:

\ln \left|x+\sqrt{x^{2}-a^{2}}\right|+C^{\prime}

\text { where } C^{\prime}=C-\ln a

 

Given:

\int \frac{1}{\sqrt{x^{2}-a^{2}}} d x

Hint:

In this statement we have to assume x as asecθ.

Solution: 

I=\int \frac{1}{\sqrt{x^{2}-a^{2}}} d x

\text { putting } x=a \sec \theta

\Rightarrow d x=a \sec \theta \tan \theta d \theta

I=\int \frac{a \sec \theta \tan \theta d \theta}{\sqrt{a^{2} \sec ^{2} \theta-a^{2}}}

=\int \frac{a \sec \theta \tan \theta d \theta}{a \cdot \tan \theta}

=\int \sec \theta \tan \theta d \theta

=\ln |\sec \theta+\tan \theta|+C

=\ln \left|\sec \theta+\sqrt{\sec ^{2} \theta-1}\right|+C

=\ln \left|\frac{x}{a}+\sqrt{\left(\frac{x}{a}\right)^{2}}-1\right|+C

=\ln \left|\frac{x+\sqrt{x^{2}-a^{2}}}{a}\right|+C

=\ln \left|x+\sqrt{x^{2}-a^{2}}\right|+C

=\ln \left|x+\sqrt{x^{2}-a^{2}}\right|-\ln a+C

=\ln \left|x+\sqrt{x^{2}-a^{2}}\right|+C^{\prime}

\text { where } C^{\prime}=C-\ln a

 

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