Evaluate the Integral

Problem

(∫_^)(x√(,x2+4)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function x2+4 is a multiple of the outer factor x

2. Substitute u=x2+4

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

4. Rewrite the integral in terms of u

(∫_^)(√(,u)⋅1/2*d(u))

5. Factor out the constant:

1/2*(∫_^)(u(1/2)*d(u))

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

1/2⋅(u(3/2))/(3/2)+C

7. Simplify the expression:

1/2⋅2/3*u(3/2)+C

1/3*u(3/2)+C

8. Back-substitute x2+4 for u to return to the original variable.

Final Answer

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

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