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Step 1: Rewrite the integrand using cot(x)
Observe that
$ \sec^{\tfrac{2}{3}}(x) = \cos^{-\tfrac{2}{3}}(x) $
and
$ \csc^{\tfrac{4}{3}}(x) = \sin^{-\tfrac{4}{3}}(x). $
We can combine them as follows:
$$
\sec^{\tfrac{2}{3}}(x)\,\csc^{\tfrac{4}{3}}(x)
\;=\; \frac{\sec^{\tfrac{2}{3}}(x)}{\csc^{\tfrac{2}{3}}(x)} \,\csc^2(x).
$$
Since
$ \csc^{\tfrac{2}{3}}(x) = \bigl(\sin^{-\tfrac{1}{3}}(x)\bigr)^{2}
= \sin^{-\tfrac{2}{3}}(x), $
we see that dividing
$ \sec^{\tfrac{2}{3}}(x) $
by
$ \csc^{\tfrac{2}{3}}(x) $
effectively becomes
$ 1 / \cot^{\tfrac{2}{3}}(x). $
So the integral becomes
$$
\int \sec^{\tfrac{2}{3}}(x)\,\csc^{\tfrac{4}{3}}(x)\,dx
\;=\;
\int \frac{1}{\cot^{\tfrac{2}{3}}(x)}\,\csc^2(x)\,dx.
$$
Step 2: Use the substitution cot(x) = t³
Let
$ \cot(x) = t^3. $
Then differentiate both sides with respect to x:
$$
\frac{d}{dx}\bigl[\cot(x)\bigr]
\;=\;
\frac{d}{dx}\bigl[t^3\bigr].
$$
We know
$ \frac{d}{dx}[\cot(x)] = -\csc^2(x), $
and
$ \frac{d}{dx}[t^3] = 3t^2\,\frac{dt}{dx}. $
Hence,
$$
-\csc^2(x)
\;=\;
3t^2 \,\frac{dt}{dx}
\quad\Longrightarrow\quad
\csc^2(x)\,dx
\;=\;
-\,3\,t^2\,dt.
$$
Step 3: Convert the integral entirely in terms of t
Substituting
$ \csc^2(x)\,dx = -\,3\,t^2\,dt $
and
$ \cot^{\tfrac{2}{3}}(x) = (t^3)^{\tfrac{2}{3}} = t^2, $
into
$ \int \frac{1}{\cot^{\tfrac{2}{3}}(x)}\,\csc^2(x)\,dx, $
we get
$$
\int \frac{1}{t^2} \;\bigl(\csc^2(x)\,dx\bigr)
\;=\;
\int \frac{1}{t^2} \,\bigl(-3\,t^2\,dt\bigr)
\;=\;
-3 \int \frac{t^2}{t^2}\,dt
\;=\;
-3\int 1\,dt.
$$
Hence,
$$
=\; -3\,t + C.
$$
Step 4: Re-substitute t = (cot(x))^(1/3) and express in terms of tan(x)
Recall
$ t = ( \cot(x) )^{\tfrac{1}{3}}. $
Therefore,
$$
-3\,t
\;=\;
-3\,(\cot(x))^{\tfrac{1}{3}}
\;=\;
-3\,\bigl(\tan(x)\bigr)^{-\tfrac{1}{3}}.
$$
So the integral evaluates to
$$
\int \sec^{\tfrac{2}{3}}(x)\,\csc^{\tfrac{4}{3}}(x)\,dx
\;=\;
-3\,\bigl(\tan(x)\bigr)^{-\tfrac{1}{3}} + C.
$$
Important Note on the sign
Checking by differentiation confirms that
$ -3\,(\tan(x))^{-\tfrac{1}{3}}
$
indeed gives
$ \sec^{\tfrac{2}{3}}(x)\,\csc^{\tfrac{4}{3}}(x)
$
when differentiated. This matches Option (4) in the provided list.
However, the question's stated "Correct Answer" is
3 tan–1/3(x) + C,
which, upon differentiation, would yield the negative
of the given integrand. Thus there appears to be a sign mismatch
in the question statement.
Mathematically, the consistent antiderivative is
$ -3\,(\tan x)^{-\tfrac{1}{3}} + C. $
