Evaluate the Integral

Problem

(∫_^)(x2√(,x+1)*d(x))

Solution

1. Substitute a new variable to simplify the radical by letting u=x+1

2. Differentiate the substitution to find d(u)=d(x)

3. Express the x terms in the integrand in terms of u by using x=u−1

4. Substitute these into the integral to get (∫_^)((u−1)2√(,u)*d(u))

5. Expand the squared binomial (u−1)2=u2−2*u+1

6. Distribute the √(,u) (which is u(1/2) across the terms to get (∫_^)((u(5/2)−2*u(3/2)+u(1/2))*d(u))

7. Integrate each term using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1)

8. Simplify the coefficients to get 2/7*u(7/2)−4/5*u(5/2)+2/3*u(3/2)+C

9. Back-substitute u=x+1 to return to the original variable.

Final Answer

(∫_^)(x2√(,x+1)*d(x))=2/7*(x+1)(7/2)−4/5*(x+1)(5/2)+2/3*(x+1)(3/2)+C

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