Question

Simplify: \[\int {\dfrac{{{{({x^4} - x)}^{1/4}}}}{{{x^5}}}} dx\]

Answer 1

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 \[{x^n} = n{x^{n - 1}}dx\].

Complete step by step answer:
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.