Calculus I (AB) and Calculus II (BC): U-substitution
(Integration by Change of Variables)
Integrate the following expressions using u-substitution:
Difficulty:
Solution:
(a) If you set
denominator of the problem to get this:
. Therefore, you can completely eliminate the
(b) If you rewrite the integral as
, it's easier to see that the derivative of tan x is
also in the problem hiding. You can set u = tan x. There is another way to do the problem that is
just as easy; if you set
favor of du:
, and the whole problem disappears in
(c) It's usually a good idea to set u equal to the entire exponential function, rather than just its
power, as we saw in (b). Here, we set
. When you take the derivative (and find du) of an
exponential function with a constant base, you first leave the function alone, then multiply it by
the derivative of the power (which is 2 in this case) and then multiply it by the natural log of the
base (here the base is 5):
Now the material on the left will replace the material on the right (as it appears in the original
integral) and you can integrate: