Evaluate:

∫ sin 2 x ⋅ cos 3 x d x

To evaluate the integral ∫ sin(2x) ⋅ cos(3x) dx, we can use the product-to-sum identities from trigonometry. This approach simplifies the expression, making it easier to integrate.

Applying the Product-to-Sum Identity

The product-to-sum identity states that:

• sin(A) cos(B) = 1/2 [sin(A + B) + sin(A - B)]

In our case, let A = 2x and B = 3x. Thus, we can rewrite the integral as:

∫ sin(2x) cos(3x) dx = ∫ (1/2) [sin(2x + 3x) + sin(2x - 3x)] dx

Simplifying the Integral

This simplifies to:

∫ (1/2) [sin(5x) + sin(-x)] dx

Since sin(-x) = -sin(x), we can further simplify:

∫ (1/2) [sin(5x) - sin(x)] dx

Integrating Each Term

Now, we can integrate each term separately:

• ∫ sin(5x) dx = -1/5 cos(5x)
• ∫ sin(x) dx = -cos(x)

Putting it all together, we have:

∫ sin(2x) cos(3x) dx = (1/2) [-1/5 cos(5x) + cos(x)] + C

Final Result

Thus, the evaluated integral is:

∫ sin(2x) cos(3x) dx = -1/10 cos(5x) + (1/2) cos(x) + C