Evaluate the Integral

Problem

(∫_^)(x5*cos(x3)*d(x))

Solution

1. Substitute u=x3 to simplify the argument of the cosine function.

2. Calculate the differential d(u)=3*x2*d(x) which implies x2*d(x)=1/3*d(u)

3. Rewrite the integral by splitting x5 into x3⋅x2 and substituting the variables.

(∫_^)(x3*cos(x3)⋅x2*d(x))=(∫_^)(u*cos(u)⋅1/3*d(u))

4. Factor out the constant to prepare for integration by parts.

1/3*(∫_^)(u*cos(u)*d(u))

5. Apply integration by parts using the formula (∫_^)(w*d(v))=w*v−(∫_^)(v*d(w)) Let w=u and d(v)=cos(u)*d(u)

6. Determine the components d(w)=d(u) and v=sin(u)

7. Evaluate the parts using the formula.

1/3*(u*sin(u)−(∫_^)(sin(u)*d(u)))

8. Integrate the remaining term noting that (∫_^)(sin(u)*d(u))=−cos(u)

1/3*(u*sin(u)+cos(u))+C

9. Back-substitute u=x3 to return to the original variable x

1/3*(x3*sin(x3)+cos(x3))+C

Final Answer

(∫_^)(x5*cos(x3)*d(x))=1/3*x3*sin(x3)+1/3*cos(x3)+C

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