∫ x 2 − 2 x 3 √ x 2 − 1 d x is equal to Options: A . &nbsp x 2 √ x 2 − 1 B . &nbsp − x 2 √ x 2 − 1 C . &nbsp √ x 2 − 1 x 2 D . &nbsp − √ x 2 − 1 x 2

Answer: Option D : D ∫ x 2 − 2 x 3 √ x 2 − 1 d x = ∫ d x x √ x 2 − 1 − 2 ∫ d x x 3 √ x 2 − 1 = s e c − 1 x − 2 ∫ s e c θ t a n θ s e c 3 θ t a n θ d θ = [ p u t x = s e c θ ⇒ d x = s e c θ t a n θ d θ ] = s e c − 1 x − 2 ∫ c o s 2 θ d θ = s e c − 1 x − 2 ∫ ( 1 + c o s 2 θ ) d θ = s e c − 1 x − ( θ + s i n 2 θ 2 ) + C = s e c − 1 x − s e c − 1 x − √ x 2 − 1 x 2 + C = − √ x 2 − 1 x 2 + C

∫x2−2x3√x2−1dx is equal to

Discussion Forum : Indefinite Integration

Question -

$\int \frac{x^{2} - 2}{x^{3} \sqrt{x^{2} - 1}} d x$ is equal to

Options:

A .

\frac{x^{2}}{\sqrt{x^{2} - 1}}

B .

- \frac{x^{2}}{\sqrt{x^{2} - 1}}

C .

\frac{\sqrt{x^{2} - 1}}{x^{2}}

D .

- \frac{\sqrt{x^{2} - 1}}{x^{2}}

Answer: Option D
:
D

\int \frac{x^{2} - 2}{x^{3} \sqrt{x^{2} - 1}} d x = \int \frac{d x}{x \sqrt{x^{2} - 1}} - 2 \int \frac{d x}{x^{3} \sqrt{x^{2} - 1}} = s e c^{- 1} x - 2 \int \frac{s e c θ t a n θ}{s e c^{3} θ t a n θ} d θ = [p u t x = s e c θ \Rightarrow d x = s e c θ t a n θ d θ] = s e c^{- 1} x - 2 \int c o s^{2} θ d θ = s e c^{- 1} x - 2 \int (1 + c o s 2 θ) d θ = s e c^{- 1} x - (θ + \frac{s i n 2 θ}{2}) + C = s e c^{- 1} x - s e c^{- 1} x - \frac{\sqrt{x^{2} - 1}}{x^{2}} + C = - \frac{\sqrt{x^{2} - 1}}{x^{2}} + C

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Discussion Forum : Indefinite Integration

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