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

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.