You are on the right track meaning that you have solved the equation almost correctly. However the last step (adding the integration constant) is wrong. The following:
is not right, you must add one constant and not two. However there are several possibilities, p.e.
As you can see the latter two are derived from the first two. It does not matter what the constant is, it can be a positive or a negative number. Don't add zero because this leads to nowhere :-) From any of these you can apply the boundary condition and thus you are able to obtain the formula needed. If you want to add two constants, they are not the same and must be called differently as follows:
which can be rewritten to any of the cases mentioned above, so in fact you are using two different ones but mostly the step of combining them is left out and one final constant is immediately written down. Can you proceed from here in finding the required formula?