Solve by techniques of integration

Question

 Integrate the following rational function:  

 S (3x ^3 - 17 x^2 + 36x  -  35) / (x^2 – 4x + 4)  dx

Answer 1

With a rational function where the numerator is a larger degree than the denominator, just perform the division prior to integrating:

So (3x³ - 17x² + 36x - 35)/(x² - 4x + 4) is equivalent to 3x - 5 with remainder 4x - 15 which can be written as:

3x - 5 + (4x - 15)/(x² - 4x + 4)

Now, the last term, (4x - 15)/(x² - 4x + 4), can be broken apart using partial fraction techniques. First factor the denominator:

(4x - 15)/(x -2)² which means you can write this in the form of A/(x - 2) + B/(x - 2)² To determine what the "A" and the "B" are, get a common denominator by multiplying the left term by (x - 2)/(x - 2) which makes the numerator A(x - 2). The other numerator is B. This means that A(x - 2) + (B) must be the same as the real numerator "4x - 15". So Ax must be 4x, so A = 4. That means 4(x - 2) + B = 4x - 15 or -8 + B = -15 or B = -7

That means we can write this last term as 4/(x - 2) - 7/(x - 2)²

So the entire integral becomes:

∫(3x - 5 + 4/(x - 2) - 7/(x - 2)² ) dx

Now, you can use easy techniques to find the antiderivative:

(3/2)x² - 5x + 4ln(x-2) + 7/(x - 2) + C

Answer 2

See the solution by using this link:

https://i.imgur.com/pGIb5Oq.png