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Last updated date: 02nd Dec 2023
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Simplify: $\int {\dfrac{{{{({x^4} - x)}^{1/4}}}}{{{x^5}}}} dx$

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Hint: The given expression has complexity in its terms. We first try to make it to a simpler term by substituting a temporary term and then we will integrate the expression. After the integration and simplification, we have to re-substitute the temporary terms to its original terms, so that we will get the answer in the original terms as given the question.

Formula used:
Some of the integration formula which we will be using is $\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}}} + c$, where$c$ is the integration constant and some differentiation formula ${x^n} = n{x^{n - 1}}dx$.

The given expression is $\int {\dfrac{{{{({x^4} - x)}^{1/4}}}}{{{x^5}}}} dx$
Taking out the term ${x^4}$commonly from the numerator we get,
$\int {\dfrac{{{{({x^4} - x)}^{1/4}}}}{{{x^5}}}} dx = \int {\dfrac{{{{({x^4})}^{1/4}}{{\left( {1 - \dfrac{x}{{{x^4}}}} \right)}^{1/4}}}}{{{x^5}}}} dx$
After making some simplification we will have,
$= \int {\dfrac{{x{{\left( {1 - \dfrac{1}{{{x^3}}}} \right)}^{1/4}}}}{{{x^5}}}} dx$
Further simplifying the above expression, we will have
$= \int {\dfrac{{{{\left( {1 - \dfrac{1}{{{x^3}}}} \right)}^{1/4}}}}{{{x^4}}}} dx$
Now we will substitute ${\left( {1 - \dfrac{1}{{{x^3}}}} \right)^{1/4}}$as $t$, that is $t = {\left( {1 - \dfrac{1}{{{x^3}}}} \right)^{1/4}}$
Claim: $t = {\left( {1 - \dfrac{1}{{{x^3}}}} \right)^{1/4}}$
Raising power to $4$on both sides we will get,
${t^4} = 1 - \dfrac{1}{{{x^3}}}$
On differentiating with respect to $t$and $x$then we get,
$4{t^3}dt = \dfrac{3}{{{x^4}}}dx$
Simplifying this we get,
$\dfrac{{dx}}{{{x^4}}} = \dfrac{4}{3}{t^3}dt$
After substitution and using the claim we get,
$= {\int {\dfrac{4}{3}{t^3}\left( {{t^4}} \right)} ^{1/4}}dt$
Simplifying further we get,
$= \int {\dfrac{4}{3}} {t^3}tdt$
Making some simplification we get,
$= \int {\dfrac{4}{3}} {t^4}dt$
Let’s take out the coefficient part outside the integration,
$= \dfrac{4}{3}\int {{t^4}dt}$
Now it is easier to integrate the above expression, on integrating with respect to $t$ we get,
$= \dfrac{4}{3}\left( {\dfrac{{{t^5}}}{5}} \right) + c$
Now let us substitute the value of$t$,
$= \dfrac{4}{{15}}{\left( {1 - \dfrac{1}{{{x^3}}}} \right)^{5/4}} + c$,
Where, $c$ is the integration constant .
The above expression is the integrated form of the given expression.

Note: Since it is impossible to integrate a function which has complexity in its term, we have used the substitution method to make it to a simpler form (i.e., $t = {\left( {1 - \dfrac{1}{{{x^3}}}} \right)^{1/4}}$) which will be easier to integrate. After the integration and simplification, we have to re-substitute the temporary terms to its original terms, so that we will get the answer in the original terms as given the question.