2. Find $$ \int \sin ^{3} x \cos x^{3} d x $$

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To solve the integral $$ \int \sin^{3} x \cos x^{3} \, dx $$, we can use substitution and trigonometric identities. Let's break it down step by step.

### Step 1: Use a substitution
We can use the substitution $$ u = \sin x $$. Then, the derivative $$ du = \cos x \, dx $$.

### Step 2: Rewrite the integral
Now, we need to express $$ \sin^3 x $$ and $$ \cos x^3 $$ in terms of $$ u $$:
- $$ \sin^3 x = u^3 $$
- $$ \cos x = \sqrt{1 - u^2} $$ (but we need to be careful with $$ \cos x^3 $$)

However, since $$ \cos x^3 $$ is not straightforward to express in terms of $$ u $$, we will instead focus on the original integral and consider a different approach.

### Step 3: Use integration by parts
We can also consider using integration by parts, but it seems complex due to the $$ \cos x^3 $$ term.

### Step 4: Simplify the integral
Instead, let's rewrite the integral using the identity for $$ \sin^3 x $$:
$$
\sin^3 x = \sin x (1 - \cos^2 x)
$$
Thus, we can rewrite the integral as:
$$
\int \sin^3 x \cos x^3 \, dx = \int \sin x (1 - \cos^2 x) \cos x^3 \, dx
$$

### Step 5: Split the integral
This gives us two integrals:
$$
\int \sin x \cos x^3 \, dx - \int \sin x \cos^2 x \cos x^3 \, dx
$$

### Step 6: Solve the first integral
For the first integral, we can use substitution again:
Let $$ v = \cos x $$, then $$ dv = -\sin x \, dx $$. The integral becomes:
$$
-\int \cos^3 x \, dx
$$

### Step 7: Solve the second integral
The second integral can be approached similarly, but it will involve more complex terms.

### Conclusion
The integral $$ \int \sin^{3} x \cos x^{3} \, dx $$ is complex and may not have a simple closed form. It may require numerical methods or special functions for evaluation.

If you would like to proceed with numerical integration or further simplification, please let me know!
The integral $$ \int \sin^{3} x \cos x^{3} \, dx $$ is complex and may not have a simple closed-form solution. It might require numerical methods or special functions for evaluation.

2. Find \( \int \sin ^{3} x \cos x^{3} d x \)

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