Integrate sin₃x · cos₄x

To integrate the expression sin(3x) · cos(4x), we can use a trigonometric identity to simplify the process. The product-to-sum formulas are particularly useful here.

Using Product-to-Sum Formulas

The product-to-sum formula states:

• sin(A) · cos(B) = 0.5 [sin(A + B) + sin(A - B)]

In this case, let A = 3x and B = 4x. Applying the formula gives:

Applying the Formula

We have:

• sin(3x) · cos(4x) = 0.5 [sin(3x + 4x) + sin(3x - 4x)]
• sin(3x) · cos(4x) = 0.5 [sin(7x) + sin(-x)]

Since sin(-x) = -sin(x), we can rewrite this as:

• sin(3x) · cos(4x) = 0.5 [sin(7x) - sin(x)]

Integrating the Expression

Now, we can integrate the simplified expression:

Step-by-Step Integration

The integral becomes:

• ∫ sin(3x) · cos(4x) dx = ∫ 0.5 [sin(7x) - sin(x)] dx

Now, we can integrate each term separately:

• ∫ sin(7x) dx = -1/7 cos(7x)
• ∫ sin(x) dx = -cos(x)

Final Result

Putting it all together, we have:

• ∫ sin(3x) · cos(4x) dx = 0.5 [-1/7 cos(7x) + cos(x)] + C

Thus, the final answer is:

-1/14 cos(7x) + 0.5 cos(x) + C