Evaluate the Integral

Problem

(∫_^)(x2*sin(x3)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function x3 is a multiple of the outer factor x2

2. Substitute a new variable u for the inner function.

u=x3

3. Differentiate u with respect to x to find the relationship between d(u) and d(x)

d(u)/d(x)=3*x2

d(u)=3*x2*d(x)

4. Rearrange the differential equation to solve for the terms present in the integral.

1/3*d(u)=x2*d(x)

5. Rewrite the integral in terms of u by substituting the expressions found in the previous steps.

(∫_^)(sin(u)⋅1/3*d(u))

1/3*(∫_^)(sin(u)*d(u))

6. Integrate the sine function with respect to u

1/3*(−cos(u))+C

7. Back-substitute the original expression for u to return the answer in terms of x

−1/3*cos(x3)+C

Final Answer

(∫_^)(x2*sin(x3)*d(x))=−1/3*cos(x3)+C

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