**2011-2012**

Find the antiderivatives of each with respect to *x*, without using *u*-substitution (i.e., without changing variables):

(a)

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

### Solution:

(a) Start by expanding the numerator, squaring (*x* – 1) and mutliplying by (2*x* + 1). Rewrite the denominator as a term with a negative exponent, and then distribute it through the quantity.

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, distribute and find the antiderivtaive. The variables make it marginally more difficult, especially when you add 1 to the weird variable powers.