$$ \int x \cdot \int ^ { 5 } ( x ^ { 2 } ) \sin ^ { 8 } ( x ^ { 2 } ) d x $$

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To solve the integral

$$
\int x \cdot \int ^ { 5 } ( x ^ { 2 } ) \sin ^ { 8 } ( x ^ { 2 } ) d x,
$$

we first need to clarify the notation. It seems that the inner integral is defined with respect to $$ x $$ and has an upper limit of 5, but the expression $$ \int ^ { 5 } ( x ^ { 2 } ) \sin ^ { 8 } ( x ^ { 2 } ) d x $$ is not standard.

Assuming you meant to write the inner integral as

$$
\int_0^5 x^2 \sin^8(x^2) \, dx,
$$

we can denote this inner integral as $$ I $$:

$$
I = \int_0^5 x^2 \sin^8(x^2) \, dx.
$$

Now, we can rewrite the original expression as:

$$
\int x \cdot I \, dx.
$$

Since $$ I $$ is a constant with respect to $$ x $$, we can factor it out of the integral:

$$
I \int x \, dx.
$$

Now, we compute the integral $$ \int x \, dx $$:

$$
\int x \, dx = \frac{x^2}{2} + C,
$$

where $$ C $$ is the constant of integration. Therefore, we have:

$$
I \cdot \left( \frac{x^2}{2} + C \right).
$$

Finally, substituting back for $$ I $$:

$$
\int x \cdot \int_0^5 x^2 \sin^8(x^2) \, dx \, dx = \int_0^5 x^2 \sin^8(x^2) \, dx \cdot \left( \frac{x^2}{2} + C \right).
$$

This is the final expression, but to evaluate it completely, we would need to compute the value of the inner integral $$ I $$.

If you need the numerical value of $$ I $$, it would typically require numerical integration techniques or special functions, as it does not have a simple closed form.

Thus, the final answer is:

$$
\int x \cdot \int_0^5 x^2 \sin^8(x^2) \, dx \, dx = \int_0^5 x^2 \sin^8(x^2) \, dx \cdot \left( \frac{x^2}{2} + C \right).
$$
The expression simplifies to $$ \int_0^5 x^2 \sin^8(x^2) \, dx \cdot \left( \frac{x^2}{2} + C \right) $$.

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