Evaluate the Integral

Problem

(∫_^)(x2√(,x3+2)*d(x))

Solution

1. Identify the substitution method as the best approach because the derivative of the inner function x3+2 is a multiple of the outer factor x2

2. Substitute u=x3+2 to simplify the integrand.

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

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

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

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

4. Rewrite the integral in terms of u by substituting u and 1/3*d(u)

(∫_^)(√(,u)⋅1/3*d(u))

1/3*(∫_^)(u(1/2)*d(u))

5. Integrate with respect to u using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1)+C

1/3⋅(u(3/2))/(3/2)+C

1/3⋅2/3*u(3/2)+C

2/9*u(3/2)+C

6. Back-substitute the original expression x3+2 for u

2/9*(x3+2)(3/2)+C

Final Answer

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

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