( a + b )^ 5 = a^5 + 5 a^4b + 10 a^3b^2 + 10 a^2b^3 + 5ab^4 + b^5 #160; ( 2 x x 2 )^ 5 #160; #160; #160; = #160; (2x)^5 + 5(2x)^ ...

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Integration by Parts

Expand: 2/x ...

Question

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$(\frac{2}{x} - \frac{x}{2})^{5}$

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Solution

$(a + b)^{5} = a^{5} + 5 a^{4} b + 10 a^{3} b^{2} + 10 a^{2} b^{3} + 5 a b^{4} + b^{5}$

$(\frac{2}{x} - \frac{x}{2})^{5}$ = $(\frac{2}{x})^{5} + 5 (\frac{2}{x})^{4} (\frac{- x}{2}) + 10 (\frac{2}{x})^{3} (\frac{- x}{2})^{2} + 10 (\frac{2}{x})^{2} (\frac{- x}{2})^{3} + 5 (\frac{2}{x}) (\frac{- x}{2})^{4} + (\frac{- x}{2})^{5}$

= $(\frac{2}{x})^{5} - 5 (\frac{2}{x})^{4} (\frac{x}{2}) + 10 (\frac{2}{x})^{3} (\frac{x}{2})^{2} - 10 (\frac{2}{x})^{2} (\frac{x}{2})^{3} + 5 (\frac{2}{x}) (\frac{x}{2})^{4} + (\frac{- x}{2})^{5}$

= $\frac{32}{x^{5}} - \frac{40}{x^{3}} + \frac{20}{x} - 5 x + \frac{5 x^{3}}{8} - \frac{x^{5}}{32}$

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