Evaluate the Integral

Problem

(∫_^)(sin(x)*cos(2*x)*d(x))

Solution

1. Apply the double-angle identity for cos(2*x) to rewrite the integrand in terms of sin(x)

cos(2*x)=1−2*sin2(x)

2. Substitute the identity into the integral.

(∫_^)(sin(x)*(1−2*sin2(x))*d(x))

3. Distribute the sin(x) term.

(∫_^)((sin(x)−2*sin3(x))*d(x))

4. Use another identity for sin3(x) or prepare for a substitution. Alternatively, use the product-to-sum identity sin(A)*cos(B)=1/2*(sin(A+B)+sin(A−B))

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

5. Simplify the expression using the odd function property sin(−x)=−sin(x)

1/2*(∫_^)((sin(3*x)−sin(x))*d(x))

6. Integrate term by term using the rule (∫_^)(sin(a*x)*d(x))=−1/a*cos(a*x)

1/2*(−1/3*cos(3*x)−(−cos(x)))+C

7. Distribute the constant and simplify the signs.

−1/6*cos(3*x)+1/2*cos(x)+C

Final Answer

(∫_^)(sin(x)*cos(2*x)*d(x))=1/2*cos(x)−1/6*cos(3*x)+C

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