Hi /sci/. My prof presented this problem as part of our first week of homework. He wants us to solve it with substitution and integration by parts, but I don't think we can progress with substitution. Even if we take his suggestion for u to start with, it just derives into 1dx, which now leaves us with two dx's, one of which is under a root. Am I wrong, or does substitution just not work here?
I don't think you understand substitution...
if u = x+1, you need to express du as a function of dx.
Here it's easy: du = dx
then you need to substitute the function of x in the integral with a new (and equivalent) function of u.
x^2sqrt(x+1)dx = (u-1)^2 sqrt(u) du
Hadn't thought of replacing the x^2 with(u-1)^2. That's definitely a tool I'm going to remember for later... But you still can't integrate the new function, can you? Maybe with another round of replacement....
you need to substitute s=sqrt(u) and everything will work out.
don't forget to add boundaries and to modify them accordingly if you make a change of variables.
if x goes from 0 to t, u will go from 1 to t+1, and s will go from 1 to sqrt(t+1)
>Americans don't even know how to integrate rational functions with goniometric and / or square root funictions
TOP 10 MY FUCKING ASS HAHAHAHAHA
Not even once, my friends. ;)