Evaluate the Integral

Problem

(∫_^)(sin(x)*cos(x)*d(x))

Solution

1. Identify the powers of the trigonometric functions. Since the power of sin(x) is odd and positive, we can reserve one sin(x) factor for the differential and convert the remaining sin(x) into terms of cos(x)

2. Rewrite the integrand using the Pythagorean identity sin(x)=1−cos(x)

(∫_^)(sin(x)*cos(x)*sin(x)*d(x))

(∫_^)((1−cos(x))*cos(x)*sin(x)*d(x))

3. Substitute u=cos(x) which implies d(u)=−sin(x)*d(x) or −d(u)=sin(x)*d(x)

(∫_^)((1−u2)*u5*(−d(u)))

4. Distribute the terms and simplify the integral expression.

(∫_^)((u7−u5)*d(u))

5. Integrate with respect to u using the power rule.

(u8)/8−(u6)/6+C

6. Back-substitute u=cos(x) to express the final answer in terms of x

cos(x)/8−cos(x)/6+C

Final Answer

(∫_^)(sin(x)*cos(x)*d(x))=cos(x)/8−cos(x)/6+C

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