∫ Sin mx cos nx =?

Given:

Integral: ∫Sin(mx) × Cos(nx) dx

Concept Used:

The trigonometric product-to-sum formula is used here:

Sin(A) × Cos(B) = 1/2 × [Sin(A+B) + Sin(A-B)]

Calculation:

Using the formula:

Sin(mx) × Cos(nx) = 1/2 × [Sin(mx + nx) + Sin(mx - nx)]

Substituting values:

⇒ Sin(mx) × Cos(nx) = 1/2 × [Sin((m+n)x) + Sin((m-n)x)]

Comparing this result with the given options, the correct answer is:

Option 4: 1/2[Sin(m+n)x + Sin(m-n)x]

Conclusion:

∴ The solution satisfies the formula, and the correct answer is Option 4.