Evaluate the Integral

Problem

(∫_^)(x2√(,x3+1)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the expression inside the square root, x3+1 is a multiple of the x2 term outside.

2. Substitute u=x3+1

3. Differentiate u with respect to x to find d(u)=3*x2*d(x)

4. Rearrange the differential to solve for the terms in the integral: 1/3*d(u)=x2*d(x)

5. Rewrite the integral in terms of u (∫_^)(√(,u)⋅1/3*d(u))

6. Apply the power rule for integration, (∫_^)(un*d(u))=(u(n+1))/(n+1) where n=1/2

7. Evaluate the integral: 1/3⋅(u(3/2))/(3/2)+C=2/9*u(3/2)+C

8. Back-substitute x3+1 for u to return to the original variable.

Final Answer

(∫_^)(x2√(,x3+1)*d(x))=2/9*(x3+1)(3/2)+C

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