Evaluate the Integral

Problem

(∫_^)(x4*sin(x5)*d(x))

Solution

1. Identify the inner function for substitution, which is u=x5

2. Differentiate u with respect to x to find d(u)=5*x4*d(x)

3. Rearrange the differential to solve for the terms in the integral: 1/5*d(u)=x4*d(x)

4. Substitute the variables into the integral to get (∫_^)(1/5*sin(u)*d(u))

5. Integrate the sine function, noting that (∫_^)(sin(u)*d(u))=−cos(u)

6. Back-substitute x5 for u and add the constant of integration C

Final Answer

(∫_^)(x4*sin(x5)*d(x))=−cos(x5)/5+C

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