Problem: evaluate the integral from 4 to 9 of 1/x^{1/2}+4 dx
A good strategy for integration using a u substitution is to see if it's possible to write your function as a product of some "inner" function and this inner function's derivative.
Here we have . Since we have inside our function, we will probably need to have this in our "inner" function that we make u. Which means we will need its derivative, as a factor in our integrand. So by manipulating the integrand slightly...
and now it's relatively easy to see that a substitution of will work nicely. Note that when and when , and also that , then the integral becomes