Evaluate the Integral

Problem

(∫_^)(sin(x)*cos(x)*d(x))

Solution

1. Identify the integral as a candidate for usubstitution because the derivative of sin(x) is cos(x)

2. Substitute u=sin(x) which implies that d(u)=cos(x)*d(x)

3. Rewrite the integral in terms of u

(∫_^)(u*d(u))

4. Apply the power rule for integration, which states that (∫_^)(un*d(u))=(u(n+1))/(n+1)+C

(u2)/2+C

5. Back-substitute the original expression for u

sin2(x)/2+C

Final Answer

(∫_^)(sin(x)*cos(x)*d(x))=sin2(x)/2+C

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