Evaluate the Integral

Problem

(∫_^)(x2*(x3+5)9*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function x3+5 is a multiple of the outer factor x2

2. Define the variable u and its differential d(u)

u=x3+5

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

3. Solve for the terms present in the integral to substitute correctly.

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

4. Substitute the expressions into the integral to change the variable from x to u

(∫_^)(u9⋅1/3*d(u))

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

1/3⋅(u10)/10+C

(u10)/30+C

6. Back-substitute the original expression for u to get the final answer in terms of x

((x3+5)10)/30+C

Final Answer

(∫_^)(x2*(x3+5)9*d(x))=((x3+5)10)/30+C

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