Evaluate the Integral

Problem

(∫_^)(x/√(,x+9)*d(x))

Solution

1. Identify a suitable substitution to simplify the integrand, letting u=x+9

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

3. Solve for x in terms of u to substitute the numerator, giving x=u−9

4. Substitute these expressions into the integral to rewrite it in terms of u

(∫_^)((u−9)/√(,u)*d(u))

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

(∫_^)((u(1/2)−9*u(−1/2))*d(u))

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

2/3*u(3/2)−18*u(1/2)+C

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

2/3*(x+9)(3/2)−18*(x+9)(1/2)+C

8. Factor out the common term 2*(x+9)(1/2) to simplify the expression:

2/3*(x+9)(1/2)*(x+9−27)+C

2/3*(x+9)(1/2)*(x−18)+C

Final Answer

(∫_^)(x/√(,x+9)*d(x))=(2*(x−18)√(,x+9))/3+C

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