∫√(5+x^10/x^10).1/x^11dx =∫√(1+5/x^10).1/x^11dx = 1/50∫√(1+5/x^10).d ( 1+5/x^10 ) = 1/50{ ( 1+5/x^10 )^3/2/3/2}+c = 1/75 ( 1+5/x^10 )^3/2+c ...

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Integration by Substitution

∫√5+x10/x16dx...

Question

$\int \frac{\sqrt{5 + x^{10}}}{x^{16}} d x =$

\frac{- 1}{75} (1 + \frac{5}{x^{10}}) + c

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\frac{- 1}{50} {(1 + \frac{5}{x^{10}})}^{3 / 2} + c

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\frac{- 1}{75} {(1 + \frac{5}{x^{10}})}^{3 / 2} + c

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\frac{- 1}{75} {(1 + \frac{5}{x^{10}})}^{5 / 2} + c

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Solution

The correct option is C $\frac{- 1}{75} {(1 + \frac{5}{x^{10}})}^{3 / 2} + c$
$\int \sqrt{\frac{5 + x^{10}}{x^{10}}} . \frac{1}{x^{11}} d x = \int \sqrt{1 + \frac{5}{x^{10}}} . \frac{1}{x^{11}} d x = - \frac{1}{50} \int \sqrt{1 + \frac{5}{x^{10}}} . d (1 + \frac{5}{x^{10}}) = - \frac{1}{50} ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \frac{{(1 + \frac{5}{x^{10}})}^{3 / 2}}{\frac{3}{2}} ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ + c = - \frac{1}{75} {(1 + \frac{5}{x^{10}})}^{\frac{3}{2}} + c$

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