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Completing the square Calculator

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1

Solved example of completing the square

$\int\frac{1}{x^2-4x+13}dx$
2

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

$\int\frac{1}{x^2-4x+13+4-4}dx$
3

Factor the perfect square trinomial $x^2+-4x+4$

$\int\frac{1}{9+\left(x-2\right)^2}dx$
4

Solve the integral $\int\frac{1}{9+\left(x-2\right)^2}dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=x-2 \\ du=dx\end{matrix}$
5

Substituting $u$ and $dx$ in the integral and simplify

$\int\frac{1}{9+u^2}du$
6

Factor the integral's denominator by $9$

$\frac{1}{9}\int\frac{1}{1+\frac{1}{9}u^2}du$
7

Solve the integral applying the substitution $v^2=\frac{1}{9}u^2$

$\frac{1}{9}\cdot 3\int\frac{1}{1+v^2}dv$
8

Multiply $\frac{1}{9}$ times $3$

$\frac{1}{3}\int\frac{1}{1+v^2}dv$
9

Solve the integral applying the formula $\displaystyle\int\frac{x'}{x^2+a^2}dx=\frac{1}{a}\arctan\left(\frac{x}{a}\right)$

$\frac{1}{3}arctan\left(v\right)$
10

Substitute $v$ back for it's value, $\frac{1}{3}u$

$\frac{1}{3}arctan\left(\frac{1}{3}u\right)$
11

Substitute $u$ back for it's value, $x-2$

$\frac{1}{3}arctan\left(\frac{1}{3}\left(x-2\right)\right)$
12

As the integral that we are solving is an indefinite integral, when we finish we must add the constant of integration

$\frac{1}{3}arctan\left(\frac{1}{3}\left(x-2\right)\right)+C_0$

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