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

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

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Answer

$\int\frac{2}{\left(x-1\right)^2+1}dx+\frac{1}{3}\int\frac{\sqrt[3]{u}}{2-2\sqrt[3]{u}+\sqrt[3]{u^{2}}}du$

Step by step solution

Problem

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

Split the fraction $\frac{x^3+2}{2-2x+x^2}$ in two terms with same denominator

$\int\left(\frac{2}{2-2x+x^2}+\frac{x^3}{2-2x+x^2}\right)dx$
2

The integral of a sum of two or more functions is equal to the sum of their integrals

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

Solve the integral $\int\frac{x^3}{-2u\left(x-1\right)}du$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=x^3 \\ du=3x^{2}dx\end{matrix}$
4

Isolate $dx$ in the previous equation

$\frac{du}{3x^{2}}=dx$
5

Rewriting $x$ in terms of $u$

$x=\sqrt[3]{u}$
6

Substituting $u$, $dx$ and $x$ in the integral

$\int\frac{2}{2-2x+x^2}dx+\int\frac{\sqrt[3]{u}}{3\left(2-2\sqrt[3]{u}+\sqrt[3]{u^{2}}\right)}du$
7

Taking the constant out of the integral

$\int\frac{2}{2-2x+x^2}dx+\frac{1}{3}\int\frac{\sqrt[3]{u}}{2-2\sqrt[3]{u}+\sqrt[3]{u^{2}}}du$
8

Add and subtract $\displaystyle\left(\frac{b}{2a}\right)^2$

$\int\frac{2}{-1+1+2-2x+x^2}dx+\frac{1}{3}\int\frac{\sqrt[3]{u}}{2-2\sqrt[3]{u}+\sqrt[3]{u^{2}}}du$
9

Factor the perfect square trinomial $x^2+-2x+1$

$\int\frac{2}{-1+2+\left(x-1\right)^2}dx+\frac{1}{3}\int\frac{\sqrt[3]{u}}{2-2\sqrt[3]{u}+\sqrt[3]{u^{2}}}du$
10

Subtract the values $2$ and $-1$

$\int\frac{2}{\left(x-1\right)^2+1}dx+\frac{1}{3}\int\frac{\sqrt[3]{u}}{2-2\sqrt[3]{u}+\sqrt[3]{u^{2}}}du$

Answer

$\int\frac{2}{\left(x-1\right)^2+1}dx+\frac{1}{3}\int\frac{\sqrt[3]{u}}{2-2\sqrt[3]{u}+\sqrt[3]{u^{2}}}du$

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Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.62 seconds

Views:

109