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\int\frac{x^3+2}{x^2-2x+2}dx

Integral of (x^3+2)/(-2x+2+x^2)

Answer

$\ln\left|2-2x+x^2\right|+x\left(2+\frac{1}{2}x\right)+C_0$

Step-by-step explanation

Problem

$\int\frac{x^3+2}{x^2-2x+2}dx$
1

Divide $2+x^3$ by $2-2x+x^2$

$\left|\begin{matrix}x^{3} & & & 2 \\ -x^{3} & 2x^{2} & -2x & \\ & 2x^{2} & -2x & 2 \\ & -2x^{2} & 4x & -4 \\ & & 2x & -2\end{matrix}\right|\begin{matrix}x^{2}-2x+2 \\ x+2 \\ \\ \\ \\ \end{matrix}$

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Answer

$\ln\left|2-2x+x^2\right|+x\left(2+\frac{1}{2}x\right)+C_0$

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$\int\frac{x^3+2}{x^2-2x+2}dx$

Main topic:

Integration by substitution

Used formulas:

6. See formulas

Time to solve it:

~ 0.97 seconds