Evaluate the Integral

Problem

(∫_^)(sin(x)*cos(x)*d(x))

Solution

1. Identify the strategy for an integral of the form (∫_^)(sin(x)*cos(x)*d(x)) Since the power of sine is odd (m=3, we can split off one sin(x) factor and use the identity sin(x)=1−cos(x)

2. Rewrite the integral by separating one sin(x) factor.

(∫_^)(sin(x)*cos(x)*sin(x)*d(x))

3. Substitute the identity sin(x)=1−cos(x) into the expression.

(∫_^)((1−cos(x))*cos(x)*sin(x)*d(x))

4. Apply a change of variables by letting u=cos(x) Then, the derivative is d(u)=−sin(x)*d(x) which means sin(x)*d(x)=−d(u)

(∫_^)((1−u2)*u4*(−d(u)))

5. Distribute the terms and simplify the integrand.

(∫_^)((u6−u4)*d(u))

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

(u7)/7−(u5)/5+C

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

cos(x)/7−cos(x)/5+C

Final Answer

(∫_^)(sin(x)*cos(x)*d(x))=cos(x)/7−cos(x)/5+C

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