Question

How do you evaluate the integral of `int ( (2x^4 + 5x^3 + 17x^2 - 42x + 19) / (x^3 + 2x^2 + 5x - 26) ) dx`?

Answer 1

`=x^2+x+23/25ln| x-2|-102/25ln|3/sqrt(x^2+4x+13)|-41/5tan^(-1)((x+2)/3)+const.`

Explanation:

First we should get in the numerator a polynomial of a grade inferior than the denominator's

By long division:

`" "2x^4+5x^3+0x^2+2x-1" "`|`" "x^3+2x^2+5x-26`
`-2x^4-4x^3-10x^2+26x" "`|____
_____#" "2x+1#
`" "x^3+7x^2-16x+19`
`" "-x^3-2x^2-5x+26`
`" "` _______
`" "5x^2-21x+45`

So the expression becomes
`=int(2x+1)dx+int (5x^2-21x+45)/(x^3+2x^2+5x-26)dx`

Let's deal with the last part
Trying `+-1, +-2 and +-3` we discover that `x=2` is a root of the polynomial of the denominator
By long division:

`" "x^3+2x^2+5x-26" "`|`" "x-2`
`-x^3+2x^2" "`|____
___#" "x^2+4x+13#
`" "4x^2+5x`
`" "-4x^2+8x`
`" "` ___
`" "13x-26`
`" "-13x+26`
`" "` ______
`" "0`

So we can write the last integrand as
`(5x^2-21x+45)/((x-2)(x^2+4x+13))=A/(x-2)+(Bx+C)/(x^2+4x+13)`

For `x=0, 1 and 3` we get

`-45/26=A/-2+C/13`
`-29/18=-A+(B+C)/18`
`27/34=A+(3B+C)/34`

Or
#[[-1/2,0,1/13],[-1,1/18,1/18],[1, 3/34,1/34]][[A],[B],[C]]=[[-45/26],[-29/18],[27/34]]#
Solving this system of variables, we get

`A=.92=23/25`
`B=4.08=102/25`
`C=-16.52=-413/25`

Then the original expression becomes
`=x^2+x+23/25int(x-2)dx+1/25int(102x-413)/(x^2+4x+13)dx`

Let's deal with the last part
`int(102x-413)/(x^2+4x+13)dx=`

Since `(x+2)^2=x^2+4x+4`
`(x+2)=3tany`
`dx=3sec^2y*dy`
How many units of `(x+2)` are there in the numerator?
`-> (102x-413)/(x+2)=102-615/(x+2)`
So the last partial expression becomes
`=102int (3tany.3cancel(sec^2 y))/(9cancel(sec^2y))dy-615int (3cancel(sec^2 y))/(9cancel(sec^y))dy`
`=-102ln|cosy|-205y`
But `sin y=(x+2)/3cosy`
And `sin^2y+cos^2y=1` => `((x^2+4x+4)/9+1)cos^2 y=1` => `cosy=3/sqrt(x^2+4x+13)`
So the partial expression becomes
`=-102ln|3/sqrt(x^2+4x+13)|-205tan^(-1)((x+2)/3)`

Finally, back to the main expression, we get
`=x^2+x+23/25ln| x-2|+1/25(-102ln|3/sqrt(x^2+4x+13)|-205tan^(-1)((x+2)/3))`
`=x^2+x+23/25ln| x-2|-102/25ln|3/sqrt(x^2+4x+13)|-41/5tan^(-1)((x+2)/3)+const.`