Mathematics

# Evaluate :$\int \dfrac{(x^{4}-x)^{1/4}}{x^{5}}dx$

##### SOLUTION
$\int \dfrac{(x^{4}-x)^{1/4}}{x^{5}}dx$
$= \int \frac{(x^{4})^{1/4}(1-\frac{x}{x^{4}})^{1/4}}{x^{5}}.dx$
$= \int \frac{x(1-\frac{1}{x^{3}})^{1/4}}{x^{5}}.dx$
$= \int \frac{(1-\frac{1}{x^{3}})^{1/4}}{x^{4}}. dx$
Let $1-\frac{1}{x^{3}} = t$
$-(\frac{-3}{x^{4}}) = \frac{dt}{dx}$
$\frac{3}{x^{4}} = \frac{dt}{dx}$
$\frac{dt}{3}= \frac{dx}{x^{4}}$
$\rightarrow = \int t^{2/4} (\frac{dt}{3})$
$= \frac{1}{3}\int t^{\frac{1}{4}}.dt$
$= \frac{1}{3}(\frac{t^{5/4}}{5/4})+C$
$= \frac{1\times 4}{3\times 5}t^{5/4}+C$
$= \frac{4}{15}t^{5/4}+C$
$= \frac{4}{15}\times (1-\frac{1}{x^{3}})^{5/4}+C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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