Evaluate the Integral

Problem

(∫_^)(sin(3*x)*cos(2*x)*d(x))

Solution

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

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

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

3. Simplify the arguments of the sine functions.

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

4. Rewrite the integral using the simplified expression and pull the constant factor outside.

(∫_^)(sin(3*x)*cos(2*x)*d(x))=1/2*(∫_^)((sin(5*x)+sin(x))*d(x))

5. Integrate each term individually using the rule (∫_^)(sin(a*x)*d(x))=−cos(a*x)/a

(∫_^)(sin(5*x)*d(x))=−cos(5*x)/5

(∫_^)(sin(x)*d(x))=−cos(x)

6. Combine the results and add the constant of integration C

1/2*(−cos(5*x)/5−cos(x))+C

7. Distribute the constant factor to reach the final form.

−cos(5*x)/10−cos(x)/2+C

Final Answer

(∫_^)(sin(3*x)*cos(2*x)*d(x))=−cos(5*x)/10−cos(x)/2+C

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