Question

If \[\int {\dfrac{{dx}}{{{x^3}{{(1 + {x^6})}^{\dfrac{2}{3}}}}}} = f(x){(1 + {x^{ - 6}})^{\dfrac{1}{3}}} + C\], C is a constant of integration, then \[f(x)\] is equal to
A. \[ - \dfrac{1}{2}\]
B. \[ - \dfrac{1}{6}\]
C. \[ - \dfrac{6}{x}\]
D. \[ - \dfrac{x}{2}\]

Answer 1

Hint: We have to choose the correct option which is equal to \[f(x)\]. They give the relation of \[f(x)\]. We find the value of \[f(x)\], at first, we will integrate the given integration.
For integration, we will apply a substitution method.
According to the substitution method, a given integral \[\int {f(x)} dx\] can be transformed into another form by changing the independent variable \[x\] to \[t\].
Then comparing the given problem we can find the value of \[f(x)\].

Complete step-by-step answer:
It is given that; \[\int {\dfrac{{dx}}{{{x^3}{{(1 + {x^6})}^{\dfrac{2}{3}}}}}} = f(x){(1 + {x^{ - 6}})^{\dfrac{1}{3}}} + C\], C is a constant of integration
We have to find the value of \[f(x)\].
\[ \Rightarrow \int {\dfrac{{dx}}{{{x^3}{{(1 + {x^6})}^{\dfrac{2}{3}}}}}} \]
The given integration can be written as,
\[ \Rightarrow \int {\dfrac{{dx}}{{{x^3}.{x^4}{{(1 + {x^{ - 6}})}^{\dfrac{2}{3}}}}}} \]
We know that, by property of index, \[{a^m}.{a^n} = {a^{m + n}}\]
So, we have,
\[ \Rightarrow \int {\dfrac{{dx}}{{{x^7}{{(1 + {x^{ - 6}})}^{\dfrac{2}{3}}}}}} \ldots {\text{ }}\left( 1 \right)\]
Now we apply a substitution.
Let us take,
\[ \Rightarrow {(1 + {x^{ - 6}})^{\dfrac{1}{3}}} = t\]
Differentiate with respect to \[x\], we get,
\[ \Rightarrow \dfrac{1}{3}{(1 + {x^{ - 6}})^{\dfrac{{ - 2}}{3}}}.( - 6{x^{ - 7}})dx = dt\]
Simplifying we get,
\[ \Rightarrow {(1 + {x^{ - 6}})^{\dfrac{{ - 2}}{3}}}.( - 2{x^{ - 7}})dx = dt\]
Simplifying again we get,
\[ \Rightarrow {(1 + {x^{ - 6}})^{\dfrac{{ - 2}}{3}}}.{x^{ - 7}}dx = - \dfrac{1}{2}dt\]
Now, substitute this value in (1) we get,
\[ \Rightarrow \int {\dfrac{{ - 1}}{2}} dt\]
Integrating we get,
\[ \Rightarrow \dfrac{{ - t}}{2} + C\]
Substitute \[{(1 + {x^{ - 6}})^{\dfrac{1}{3}}} = t\] in the above equation we get,
\[ \Rightarrow \dfrac{{ - 1}}{2}{(1 + {x^{ - 6}})^{\dfrac{1}{3}}} + C\]
Comparing with the given question we get,
\[f(x) = \dfrac{{ - 1}}{2}\]
$\therefore $ Hence, the correct answer is \[ - \dfrac{1}{2}\].

So, the correct answer is “Option A”.

Note: According to the substitution method, a given integral \[\int {f(x){\text{ }}dx} \] can be transformed into another form by changing the independent variable \[x\] to \[t\].
It is important to note here that you should make the substitution for a function whose derivative also occurs in the integrand.
Integrating constant is added to the function obtained by evaluating the indefinite integral of a given function, indicating that all indefinite integrals of the given function differ by, at most, a constant.