Evaluate the Integral

Problem

(∫_^)(2*x√(,x2+1)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function x2+1 is 2*x which is present in the integrand.

2. Substitute u=x2+1

3. Differentiate u with respect to x to find d(u)=2*x*d(x)

4. Rewrite the integral in terms of u

(∫_^)(√(,u)*d(u))

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

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

6. Simplify the coefficient:

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

7. Back-substitute x2+1 for u to return to the original variable.

Final Answer

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

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