Evaluate the Integral

Problem

(∫_^)(x/√(,x+4)*d(x))

Solution

1. Identify the substitution to simplify the radical in the denominator. Let u=x+4

2. Differentiate the substitution to find the relationship between d(u) and d(x)

d(u)=d(x)

3. Solve for x in terms of u to substitute into the numerator.

x=u−4

4. Substitute the expressions for x √(,x+4) and d(x) into the integral.

(∫_^)((u−4)/√(,u)*d(u))

5. Distribute the denominator to split the integrand into two separate terms.

(∫_^)((u(1/2)−4*u(−1/2))*d(u))

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

(u(3/2))/(3/2)−4(u(1/2))/(1/2)+C

7. Simplify the coefficients of the resulting expression.

2/3*u(3/2)−8*u(1/2)+C

8. Back-substitute u=x+4 to return to the original variable x

2/3*(x+4)(3/2)−8*(x+4)(1/2)+C

Final Answer

(∫_^)(x/√(,x+4)*d(x))=2/3*(x+4)(3/2)−8√(,x+4)+C

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