Calculus 2 Integration

Question

I have more problem on my webwork that is confusing me:

 

Make the substitution u=√x to express the integrand as a rational function and then evaluate the integral ∫(6√x)/(x+9)dx

Answer 1

u = √x

u² = x

 

 

6∫u(u² + 9)^-1 du

     

 

Let          r = u² + 9

              dr = 2udu

             (1/2)dr = udu

 

 

Now we can make the substitution.

 

6∫r^-1(1/2)dr =

 

3∫r^-1ru =

 

3ln(r) + C =

  

3ln(u² + 9) + C

 

 

To check, we take the derivative of this answer.

 

3 * (2u / (u² + 9)) =

 

6u / (u² + 9)

 

 

Then we substitute u=√x.

 

(6√x) / (x + 9)

 

 

We get back the original problem.