Problem 14: Power Rule for Integration

Photo by Charleston's TheDigitel
Photo by Charleston’s TheDigitel

2011-2012

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

(a) p14im1

(b) p14im2, assuming a, b, c, d, and e are constants


Solution:

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

p14im3

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.

p14im4

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.

p14im5

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