x^3(2+x^2)^1/3=x^3.(2+x^3)^ 13 #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; =(2)^ 1/3.x^ ...

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Introducing Epsilon

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Question

Expand $\frac{x^{3}}{(2 + x^{2})^{\frac{1}{3}}}$

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Solution

$\frac{x^{3}}{(2 + x^{2})^{1 / 3}} = x^{3} . (2 + x^{3})^{\frac{- 1}{3}}$

$= (2)^{- 1 / 3} . x^{3} (1 + \frac{x^{2}}{2})^{\frac{- 1}{3}}$

$= \frac{x^{3}}{2^{1 / 3}} (1 + \frac{x^{2}}{2})^{\frac{- 1}{3}}$

$= \frac{x^{3}}{2^{\frac{1}{3}}} [1 - \frac{1}{3} . (\frac{x^{2}}{2}) + \frac{\frac{1}{3} \frac{4}{3}}{2!} . (\frac{x^{2}}{2})^{2} - \frac{\frac{1}{3} \frac{4}{3} \frac{7}{3}}{3!} . (\frac{x^{2}}{2})^{3} + . . . . . . . . . . .]$

$= \frac{x^{3}}{2^{\frac{1}{3}}} [1 - \frac{x^{2}}{6} + \frac{x^{4}}{18} - \frac{7 x^{6}}{3^{4} {.2}^{2}} + . . . . . . . . . . . . . . .]$

Formula
$∵ (1 + x)^{- \frac{p}{q}} = 1 - \frac{\frac{p}{q}}{1!} . x + \frac{\frac{p}{q} \frac{(p + q)}{q}}{2!} . x^{2} - \frac{\frac{p}{q} \frac{(p + q)}{q} \frac{(p + 2 q)}{q}}{3!} . x^{3} + . . . . . . . . . . .$

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Similar questions

lim x \to \infty ((x^{3} + x^{2})^{1 / 3} - (x^{3} - x^{2})^{1 / 3}) = R

3 R =

(x^{2})^{\frac{1}{2}} \times x^{3} =

(1 + x + x^{2}) (1 - \frac{x}{1!} + \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + \dots) = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + . . .

\int e^{x} [\frac{x^{3} + x + 1}{(1 + x^{2})^{3 / 2}}] d x

\frac{1 - 2 x + x^{2}}{1 - x^{3}} \times \frac{1 + x + x^{2}}{1 + x}

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