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Step-by-Step Solution
Step 1: Recognize the Integral
We are given the integral
$ \displaystyle \int \frac{dx}{x^3(1 + x^6)^{\tfrac{2}{3}}} $
and the result in the form
$ x\, f(x)\,(1 + x^6)^{\tfrac{1}{3}} + C. $
We need to determine the function $f(x)$.
Step 2: Rewrite the Integral
Observe that $1 + x^6$ in the denominator is raised to the power of $\tfrac{2}{3}$. A common strategy is to look for a substitution that simplifies this expression. We start by expressing it in a way that might facilitate a substitution:
$ \displaystyle I \;=\; \int \frac{dx}{x^3 (1 + x^6)^{\tfrac{2}{3}}}.
$
Step 3: Use a Substitution
Let
$ \displaystyle t = \frac{1}{x^6} + 1. $
Differentiate both sides with respect to $x$:
$ \displaystyle
t = \frac{1}{x^6} + 1
\quad \Longrightarrow \quad
dt = -\frac{6}{x^7}\, dx.
$
Hence,
$ \displaystyle
\frac{dx}{x^7} = -\frac{dt}{6}.
$
We also adjust the integral accordingly. Notice that $x^3$ in the denominator can be combined with $x^4$ (from $x^7$) if needed. More directly, rewrite:
$ \displaystyle
\frac{1}{x^3 (1 + x^6)^{\tfrac{2}{3}}}
= \frac{1}{x^7 \,\bigl(\frac{1}{x^6}+1\bigr)^{\tfrac{2}{3}}}.
$
Step 4: Substitute Into the Integral
With
$ \displaystyle
t = \frac{1}{x^6} + 1,
$
and
$ \displaystyle
\frac{dx}{x^7} = -\frac{dt}{6},
$
the integral becomes:
$ \displaystyle
I \;=\; \int \frac{dx}{x^7} \,\frac{1}{\bigl(\tfrac{1}{x^6}+1\bigr)^{\tfrac{2}{3}}}
= \int \frac{1}{t^{\tfrac{2}{3}}} \;\left(-\tfrac{dt}{6}\right).
$
$ \displaystyle
I = -\tfrac{1}{6} \int t^{-\tfrac{2}{3}} \, dt.
$
Step 5: Integrate with Respect to t
$ \displaystyle
I = -\tfrac{1}{6} \int t^{-\tfrac{2}{3}} \, dt
= -\tfrac{1}{6} \left[ \frac{t^{\tfrac{1}{3}}}{\tfrac{1}{3}} \right] + C
= -\tfrac{1}{6} \bigl( 3\,t^{\tfrac{1}{3}} \bigr) + C
= -\tfrac{1}{2}\, t^{\tfrac{1}{3}} + C.
$
Step 6: Substitute Back for t
Recall that
$ \displaystyle
t = \frac{1}{x^6} + 1.
$
Thus,
$ \displaystyle
I = -\tfrac{1}{2}\,\Bigl(\frac{1}{x^6} + 1\Bigr)^{\tfrac{1}{3}} + C.
$
We can rewrite this to match the form
$ x\, f(x)\, (1 + x^6)^{\tfrac{1}{3}} + C
$ by factoring out $x\,(1 + x^6)^{\tfrac{1}{3}}$ appropriately:
$ \displaystyle
-\tfrac{1}{2}\,\Bigl(1 + x^6\Bigr)^{\tfrac{1}{3}} \frac{1}{x^2}
= x \,\Bigl( -\tfrac{1}{2x^3} \Bigr) (1 + x^6)^{\tfrac{1}{3}}.
$
Step 7: Identify f(x)
We match this to
$ \displaystyle
x \,f(x)\,(1 + x^6)^{\tfrac{1}{3}}.
$
Clearly,
$ \displaystyle
f(x) = -\tfrac{1}{2x^3}.
$
Answer
$ \displaystyle f(x) = -\frac{1}{2x^3}.
