Calculus How To

Integration Using Long Division: Definition, Examples

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Integration Using Long Division works best for rational expressions where the degree of the polynomial in the numerator is greater than or equal to the degree of the polynomial in the denominator.

Integration Using Long Division: Examples

Example question #1: Solve the following integral:
long division integral example

Step 1: Check to see if the degree in the numerator is greater than or equal to the degree in the denominator. For this example, it is, so we can continue.

Step 2: Divide the numerator by the denominator, using algebra:
long division integrals 3

Step 3: Rewrite the integral in the question with the new expression you got in Step 2:
integration by long division

Step 4: Solve the integral using the usual rules of integration:

  1. Apply the sum rule:
  2. Apply the common integral1x dx = ln |x| + c: = 2x + 3 ln |x – 2|
  3. Add a constant: 2x + 3 ln |x – 2| + C

Note: The sum rule states that the integral of the sum of two functions is the sum of their separate integrals:
∫[f(x) + g(x)] dx = ∫f(x)dx + ∫g(x) dx

Example question #2: Solve the following integral:


Step 1: Check to see if the degree in the numerator is greater than or equal to the degree in the denominator. For this example, it is, so we can continue.


Step 2: Divide the numerator by the denominator, using algebra:

We can’t actually divide this until we factor the denominator first:
step 1 factoring

Now we can perform the long division, dividing the numerator by the denominator:

The x – 1 is the quotient and 3x -2 is the remainder. Rewrite the solution above as “quotient + remainder/original factored denominator”:
rewriting

Step 3: Solve the integral using the usual rules of integration: In addition to the sum rule and common integral ∫1x dx = ln |x| + c: = 2x + 3 ln |x – 2|, we also need to apply the power rule to the “x” in front. The solution is:
integral solution

CITE THIS AS:
Stephanie Glen. "Integration Using Long Division: Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/integration-using-long-division/
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