Evaluate the Integral

Problem

(∫_^)(sin4(x)*cos(x)*d(x))

Solution

1. Identify the structure of the integrand as a power of a function multiplied by its derivative, which suggests using the substitution method.

2. Substitute u=sin(x) to simplify the integral.

3. Differentiate u with respect to x to find d(u)=cos(x)*d(x)

4. Rewrite the integral in terms of u

(∫_^)(u4*d(u))

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

(u5)/5+C

6. Back-substitute the original expression for u to return to the variable x

Final Answer

(∫_^)(sin4(x)*cos(x)*d(x))=sin5(x)/5+C

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