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

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!