If for a suitable chosen integer m and a function A(x), where C is a constant of integration, then equals:

Engineering

Mathematics

Integration of Miscellaneous Functions

Integration by Substitution

maximum and minimum of quadratic and rational expression

Question

If $\int \frac{\sqrt{1 - x^{2}}}{x^{4}} dx = A (x) {(\sqrt{1 - x^{2}})}^{m} + C$ for a suitable chosen integer m and a function A(x), where C is a constant of integration, then ${(A (x))}^{m}$ equals:

$\frac{- 1}{3 x^{3}}$

$\frac{1}{27 x^{6}}$

$\frac{- 1}{27 x^{9}}$

$\frac{1}{9 x^{4}}$

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$I = \int \frac{1}{x^{4}} \sqrt{1 - x^{2}} dx = \int \frac{1}{x^{3}} \sqrt{\frac{1}{x^{2}} - 1} dx$

$\frac{1}{x^{2}} - 1 = t^{2} \Rightarrow \frac{dx}{x^{3}} = - tdt$

$I = \int - t^{2} dt = \frac{- {(\sqrt{1 - x^{2}})}^{3}}{3 x^{3}} + C$

$\Rightarrow A (x) = - \frac{1}{3 x^{3}} and m = 3$