Evaluate the Integral

Problem

(∫_^)(sin(2*x)*cos(x)*d(x))

Solution

1. Apply the double angle identity for sine, which states sin(2*x)=2*sin(x)*cos(x) to rewrite the integrand.

(∫_^)(2*sin(x)*cos(x)*cos(x)*d(x))

2. Simplify the expression by combining the cosine terms.

(∫_^)(2*sin(x)*cos2(x)*d(x))

3. Use u-substitution by letting u=cos(x) Then, the derivative is d(u)/d(x)=−sin(x) which means d(u)=−sin(x)*d(x) or −d(u)=sin(x)*d(x)

(∫_^)(2*u2*(−d(u)))

4. Factor out the constant and integrate with respect to u using the power rule.

−2*(∫_^)(u2*d(u))

−2⋅(u3)/3+C

5. Substitute back the original expression for u where u=cos(x) to find the final result.

−2/3*cos3(x)+C

Final Answer

(∫_^)(sin(2*x)*cos(x)*d(x))=−2/3*cos3(x)+C

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