Evaluate the Integral

Problem

(∫_^)(x^2√(,1+x^3)*d(x))

Solution

1. Identify a suitable substitution to simplify the integrand, noting that the derivative of the expression inside the square root, 1+x^3 is proportional to the x^2 term outside.

2. Substitute u=1+x^3

3. Differentiate u with respect to x to find d(u)=3*x^2*d(x) which implies x^2*d(x)=1/3*d(u)

4. Rewrite the integral in terms of u

(∫_^)(√(,u)⋅1/3*d(u))

5. Factor out the constant:

1/3*(∫_^)(u^(1/2)*d(u))

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

1/3⋅(u^(3/2))/(3/2)+C

7. Simplify the constant coefficient:

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

8. Back-substitute u=1+x^3 to express the result in terms of x

Final Answer

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

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