Since the variable of integration is *x*, the limits of integration are values of *x*. The integral is read "the integral of 3(3*x* + 1)^{5} as *x* goes from 0 to 1." To be explicit, we could write the integral as If we take *u* = 3*x* + 1
*du* = 3*dx*
and start the substitution, we run into a problem: This could be read " the integral of *u*^{5} as *x* goes from 0 to 1," which doesn't make sense! The problem is that the limits of integration are values of *x*, but now the variable of integration is *u*. We can fix this by changing the limits of integration to values of *u*. We know that *u* = 3*x* + 1.
So when *x* = 0, the corresponding value of *u* is *u* = 3(0) + 1 = 1.
This lets us change the lower limit of integration from a value of *x* to a value of *u*: When *x* = 1 we have *u* = 3(1) + 1 = 4.
This lets us change the upper limit of integration: Now that we have everything in terms of *u*, there's no reason to go back to *x*. We can evaluate the integral we have, with no *x* in sight: It's reassuring that we got the same answer as when we did this problem the other way! |