Evaluate the Integral

Problem

(∫_^)(sin(3*x)*cos(5*x)*d(x))

Solution

1. Apply the product-to-sum identity for the product of sine and cosine, which is sin(A)*cos(B)=1/2*(sin(A+B)+sin(A−B))

2. Substitute A=3*x and B=5*x into the identity to rewrite the integrand.

sin(3*x)*cos(5*x)=1/2*(sin(3*x+5*x)+sin(3*x−5*x))

3. Simplify the arguments inside the sine functions.

1/2*(sin(8*x)+sin(−2*x))

4. Use the odd property of the sine function, sin(−θ)=−sin(θ) to simplify the expression further.

1/2*(sin(8*x)−sin(2*x))

5. Integrate the terms individually using the rule (∫_^)(sin(a*x)*d(x))=−1/a*cos(a*x)+C

(∫_^)(1/2*(sin(8*x)−sin(2*x))*d(x))=1/2*(−cos(8*x)/8−(−cos(2*x)/2))+C

6. Distribute the constant and simplify the signs.

−cos(8*x)/16+cos(2*x)/4+C

Final Answer

(∫_^)(sin(3*x)*cos(5*x)*d(x))=cos(2*x)/4−cos(8*x)/16+C

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