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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: