Find the antiderivatives of each of the following without using u-substitution (also called change of variables):

(a)

(b) , assuming that a, b, c, d, and e are constants

Difficulty:

Solution

(a) Start by multiplying out the numerator and rewriting the denominator as a term with a negative exponent. Then distribute:

Now you can antidifferentiate by adding 1 to each power individually and multiplying the coefficient by the reciprocal of the resulting exponent. In other words, for the first term, add 1 to 3/2 to get 5/2. Then, multiply 2 by the reciprocal of 5/2, which is 2/5:

Don't forget that a general antiderivative (i.e. an indefinite integral) must always contain "+C."

(b) Follow the same procedure, beginning by writing the radical term with a fractional exponent. Then, you'll distribute and find the antiderivtaive. The variables make it only marginally more difficult, especially when you add 1 to the weird variable powers: