{dx}{(x²-a²)³/2}} =

Correct Answer is : −xa2x2−a2+C-{x}{a²{x²-a²}}+C−a2x2−a2x+C

Correct Answer is : −xa2x2−a2+C-{x}{a²{x²-a²}}+C−a2x2−a2x+C

Test Index

TS EAMCET 6-Aug-2021 Shift 1 Question Paper

Section: Mathematics

Question : 76 of 160

Marks: +1, -0

\int \frac{dx}{(x^2-a^2)^{3/2}}

Solution:

Let

I=\int\;\frac{dx}{(x^{2}-a^{2})^{3/2}}

Putting

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

I = \int\; \frac{a \sec \theta \cdot \tan \theta \cdot d\theta}{[a^{2}(\sec^{2}\theta-1)]^{\frac{3}{2}}} = \int\; \frac{1}{a^{2}} \times \frac{\sec \theta \cdot \tan \theta}{\tan^{3} \theta^{2}} d\theta

= \frac{1}{a^{2}} \int\; \frac{1}{\cos \theta \cdot \frac{\sin^{2} \theta}{\cos^{2} \theta}} d\theta = \frac{1}{a^{2}} \int\; \frac{\cos \theta}{\sin^{2} \theta} d\theta

= \frac{1}{a^{2}} \int \cot \theta \cdot \csc \theta d\theta = -\frac{1}{a^{2}} \csc^{2} \theta + C

= -\frac{x}{a^{2} \sqrt{x^{2}-a^{2}}} + C

(\because \sec\theta = \frac{x}{a} \Rightarrow \cos\theta = \frac{a}{x} \therefore \sin\theta = \sqrt{1-\frac{a^{2}}{x^{2}}}

\Rightarrow \therefore \cos\theta = \sqrt{\frac{x^{2}}{x^{2}-a^{2}}} = \frac{x}{\sqrt{x^{2}-a^{2}}})

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