Question

Antiderivative help (will put in comment)

Answer 1

\[\int\limits_{}^{} ((3x ^{3}+2x ^{2}-12x+9) \div (x-1)) dx\]

Answer 2

use polynomial division

Answer 3

can you do long divisions of polynomials ?

Answer 4

$$\begin{align*}3x^3+2x^2-12x+9&=x^2(3x+2)-3(4x+3)\\&=x^2(3x-3+5)-3(4x-4+7)\\&=x^2(3(x-1)+5)-3(4(x-1)+7)\\&=3x^2(x-1)-12(x-1)+5x^2-21\\&=(3x^2-12)(x-1)+(5x^2-21)\end{align*}$$

Answer 5

now \(5x^2-21=5x^2-5-16=5(x^2-1)-16=(5x+5)(x-1)-16\) so $$(3x^2-12+5x+5)(x-1)-16=(3x^2+5x-7)(x-1)-16$$ and dividing thru by \(x-1\) we get:$$3x^2+5x-7-\frac{16}{x-1}$$

Answer 6

now integrate term by term:$$\int(3x^2+5x-7-16/(x-1))\ dx=x^3+\frac52x^2-7x-16\log(x-1)+C$$

Answer 7

wow, thanks!

Answer 8

wait oops there's an error here

Answer 9

found it:$$\begin{align*}3x^3+2x^2-12x+9&=x^2(3x+2)-3(4x\color{red}-3)\\&=x^2(3x-3+5)-3(4x-4\color{red}{+1})\\&=x^2(3(x-1)+5)-3(4(x-1)\color{red}{+1})\\&=3x^2(x-1)-12(x-1)+5x^2\color{red}{-3}\\&=(3x^2-12)(x-1)+(5x^2\color{red}{-3})\end{align*}$$

Answer 10

now \(5x^2\color{red}{-3}=5x^2-5\color{red}{+2}=5(x^2-1)\color{red}{+2}=(5x+5)(x-1)\color{red}{+2}\)
so $$(3x^2-12+5x+5)(x-1)\color{red}{+2}=(3x^2+5x-7)(x-1)\color{red}{+2}$$

Answer 11

dividing thru by \(x-1\) we get:$$\frac{3x^3+2x^2-12x+9}{x-1}=3x^2+5x-7\color{red}+\frac{\color{red}2}{x-1}$$aand then we integrate term by term:$$\int\left(3x^2+5x-7\color{red}+\frac{\color{red}2}{x-1}\right)\ dx=x^3+\frac52x^2-7x\color{red}{+2}\log(x-1)+C$$