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
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