Evaluate the Integral

Problem

(∫_^)(u2√(,u3+2)*d(u))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the expression inside the square root, u3+2 is a multiple of the u2 term outside.

2. Define the substitution variable w=u3+2

3. Differentiate w with respect to u to find d(w)=3*u2*d(u)

4. Rearrange the differential to solve for the terms present in the integral: 1/3*d(w)=u2*d(u)

5. Substitute the new variables into the integral to transform it into a simpler form.

(∫_^)(√(,w)⋅1/3*d(w))

6. Rewrite the square root as a fractional exponent to prepare for integration.

1/3*(∫_^)(w(1/2)*d(w))

7. Apply the power rule for integration, which states (∫_^)(wn*d(w))=(w(n+1))/(n+1)+C

1/3⋅(w(3/2))/(3/2)+C

8. Simplify the resulting expression by multiplying the fractions.

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

9. Back-substitute the original expression u3+2 in place of w to get the final result.

2/9*(u3+2)(3/2)+C

Final Answer

(∫_^)(u2√(,u3+2)*d(u))=2/9*(u3+2)(3/2)+C

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