What is $int (-x^3+9x-1 ) / (-2x^2- 3 x +5 )$ ?

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Answer 1 · 2016-12-08T11:54:11

The answer is $=x^2/4-3/4x-9/8ln(∣2x+5∣)-ln(∣x-1∣)+C$

Let's do a long division

$color(white)(aaaa)$ $-x^3+$ $color(white)(aaaaaaaaa)$ $9x-1$ $color(white)(aa)$ $∣$ $-2x^2-3x+5$

$color(white)(aaaa)$ $-x^3+$ $color(white)(a)$ $-3/2x^2+5/2x$ $color(white)(aaaaa)$ $∣$ $x/2-3/4$

$color(white)(aaaaaaa)$ $0+$ $color(white)(a)$ $+3/2x^2+13/2x-1$

$color(white)(aaaaaaaaaaa)$ #color(white)(aaa)3/2x^2+9/4x-15/4#

$color(white)(aaaaaaaaaaa)$ #color(white)(aaaaa)0+17/4x+11/4#

$-2x^2-3x+5=-(2x+5)(x-1)$

So,

$(-x^3+9x-1)/(-2^2-3x+5)=(x/2-3/4)+(17/4x+11/4)/(-2x^2-3x+5)$

$=(x/2-3/4)-(17/4x+11/4)/((2x+5)(x-1))$

Let's do a partial fraction decomposition

$(17/4x+11/4)/((2x+5)(x-1))=A/(2x+5)+B/(x-1)$

$=(A(x-1)+B(2x+5))/((2x+5)(x-1))$

$17/4x+11/4=A(x-1)+B(2x+5)$

Let $x=1$ , $=>$ , $28/4=7B$ , $=>$ , $B=1$

Let $x=0$ , $=>$ , $11/4=-A+5B$

$A=5-11/4=9/4$

So,
$int((-x^3+9x-1)dx)/(-2^2-3x+5)$

$=int(x/2-3/4)dx-9/4intdx/(2x+5)-intdx/(x-1)$

$=x^2/4-3/4x-9/8ln(∣2x+5∣)-ln(∣x-1∣)+C$

What is int (-x^3+9x-1 ) / (-2x^2- 3 x +5 )int (-x^3+9x-1 ) / (-2x^2- 3 x +5 )?