1. ## Tough integral

I encountered the following integral and I am bewildered:

$\displaystyle \int\frac{x^5-2x^4+3}{x^3-3x^2-10x}dx$

Seems to be real hard.

2. Hello, totalnewbie!

$\displaystyle \int\frac{x^5-2x^4+3}{x^3-3x^2-10x}dx$

Note that the numerator is of a higher degree than the denominator.

$\displaystyle \text{Long division: }\frac{x^5-2x^4+3}{x^3-3x^2-10x}\;=\;x^2+x+13 + \underbrace{\frac{49x^2+130x + 3}{x(x-5)(x+2)}}_{\text{use Partial Fractions}}$

3. How do you get that

$\displaystyle \text{Long division: }\frac{x^5-2x^4+3}{x^3-3x^2-10x}\;=\;x^2+x+13 + \underbrace{\frac{49x^2+130x + 3}{x(x-5)(x+2)}}_{\text{use Partial Fractions}}$

4. I'm not skilled enough at Latex to type it up for you, but have you ever done long division with polynomials? The first three terms Soroban gave you divide evenly, then the last term is the remainder.

5. Originally Posted by Jameson
I'm not skilled enough at Latex to type it up for you, but have you ever done long division with polynomials? The first three terms Soroban gave you divide evenly, then the last term is the remainder.
I have never done with polynomials. I try to probe.

6. Originally Posted by totalnewbie
How do you get that

$\displaystyle \text{Long division: }\frac{x^5-2x^4+3}{x^3-3x^2-10x}\;=\;x^2+x+13 + \underbrace{\frac{49x^2+130x + 3}{x(x-5)(x+2)}}_{\text{use Partial Fractions}}$
Code:
                 __x^2 + x + 13___________________
x^3 - 3x^2 - 10x ) x^5 - 2x^4 + 0x^3 + 0x^2 + 0x + 3
-(x^5 - 3x^4 - 10x^3)
x^4 + 10x^3 + 0x^2
-(x^4 -  3x^3 - 10x^2)
13x^3 + 10x^2 + 0x
-(13x^3 - 39x^2 -130x)
49x^2 + 130x + 3
-Dan

7. I got the right answer. Thank you all.