Final Answer
$\frac{3}{4}x^{4}+2x^{3}+\frac{7}{2}x^2+16x+31\ln\left(x-2\right)+C_0$
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Step-by-step Solution
Specify the solving method
1
Divide $3x^4-5x^2+2x-1$ by $x-2$
$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}-2;}{\phantom{;}3x^{3}+6x^{2}+7x\phantom{;}+16\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}-2\overline{\smash{)}\phantom{;}3x^{4}\phantom{-;x^n}-5x^{2}+2x\phantom{;}-1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}-2;}\underline{-3x^{4}+6x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-3x^{4}+6x^{3};}\phantom{;}6x^{3}-5x^{2}+2x\phantom{;}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n;}\underline{-6x^{3}+12x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-6x^{3}+12x^{2}-;x^n;}\phantom{;}7x^{2}+2x\phantom{;}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n;}\underline{-7x^{2}+14x\phantom{;}\phantom{-;x^n}}\\\phantom{;;-7x^{2}+14x\phantom{;}-;x^n-;x^n;}\phantom{;}16x\phantom{;}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n-;x^n;}\underline{-16x\phantom{;}+32\phantom{;}\phantom{;}}\\\phantom{;;;-16x\phantom{;}+32\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}31\phantom{;}\phantom{;}\\\end{array}$
2
Resulting polynomial
$\int\left(3x^{3}+6x^{2}+7x+16+\frac{31}{x-2}\right)dx$
3
Expand the integral $\int\left(3x^{3}+6x^{2}+7x+16+\frac{31}{x-2}\right)dx$ into $5$ integrals using the sum rule for integrals, to then solve each integral separately
$\int3x^{3}dx+\int6x^{2}dx+\int7xdx+\int16dx+\int\frac{31}{x-2}dx$
4
The integral $\int3x^{3}dx$ results in: $\frac{3}{4}x^{4}$
$\frac{3}{4}x^{4}$
5
The integral $\int6x^{2}dx$ results in: $2x^{3}$
$2x^{3}$
6
The integral $\int7xdx$ results in: $\frac{7}{2}x^2$
$\frac{7}{2}x^2$
7
The integral $\int16dx$ results in: $16x$
$16x$
8
The integral $\int\frac{31}{x-2}dx$ results in: $31\ln\left(x-2\right)$
$31\ln\left(x-2\right)$
9
Gather the results of all integrals
$\frac{3}{4}x^{4}+2x^{3}+\frac{7}{2}x^2+16x+31\ln\left(x-2\right)$
10
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
$\frac{3}{4}x^{4}+2x^{3}+\frac{7}{2}x^2+16x+31\ln\left(x-2\right)+C_0$
Final Answer
$\frac{3}{4}x^{4}+2x^{3}+\frac{7}{2}x^2+16x+31\ln\left(x-2\right)+C_0$