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  • Please Solve R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise Question 85 Maths Textbbok Solution.

Please Solve R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise  Revision Exercise Question 85 Maths Textbbok Solution.

Answers (1)

Answer:

\frac{x}{2} \sqrt{x^{2}-a^{2}}-\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}-a^{2}}\right|+c

Hint:

You must know about integration of \sqrt{x^{2}-a^{2}}

Given:

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

Solution:

I=\int 1_{I I} \cdot \sqrt{x_{I}^{2}-a^{2}} d x

=\sqrt{x^{2}-a^{2}} \int 1 d x-\int\left(\frac{d}{d x}\left(\sqrt{x^{2}-a^{2}}\right) \int 1 d x\right) d x

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

=\sqrt{x^{2}-a^{2}} \cdot x-\int\left(\frac{x^{2}-a^{2}+a^{2}}{\sqrt{x^{2}-a^{2}}}\right) d x

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

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

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

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