New Notes Blog: `int 2 / sqrt(-x^2+4x) dx` Find or evaluate the integral by completing the square

Sunday, 8 March 2015

$\displaystyle\int\frac{{2}}{\sqrt{{-{{x}}^{{2}}+{4}{x}}}}{\left.{d}{x}\right.}$ Find or evaluate the integral by completing the square

To evaluate the given integral: $\displaystyle\int\frac{{2}}{\sqrt{{-{{x}}^{{2}}+{4}{x}}}}{\left.{d}{x}\right.}$ , we may apply the basic integration property: $\displaystyle\int{c}\cdot{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}$ .

The integral becomes:

$\displaystyle{2}\int\frac{{\left.{d}{x}}}{\sqrt{{-{{x}}^{{2}}+{4}{x}}}}$

We complete the square for the expression $\displaystyle{\left(-{{x}}^{{2}}+{4}{x}\right)}$ .

Completing the square:

For the first step, factor out (-1): $\displaystyle{\left(-{{x}}^{{2}}+{4}{x}\right)}={\left(-{1}\right)}{\left({{x}}^{{2}}-{4}{x}\right)}{\quad\text{or}\quad}-{\left({{x}}^{{2}}-{4}{x}\right)}$

The $\displaystyle{{x}}^{{2}}-{4}{x}$ or $\displaystyle{{x}}^{{-{{4}}}}{x}+{0}$ resembles the $\displaystyle{a}{{x}}^{{2}}+{b}{x}+{c}$ where:

$\displaystyle{a}={1}$ , $\displaystyle{b}=-{4}$ and $\displaystyle{c}={0}$ .

To complete the square, we add...

The integral becomes:

$\displaystyle{2}\int\frac{{\left.{d}{x}}}{\sqrt{{-{{x}}^{{2}}+{4}{x}}}}$

We complete the square for the expression $\displaystyle{\left(-{{x}}^{{2}}+{4}{x}\right)}$ .

Completing the square:

For the first step, factor out (-1): $\displaystyle{\left(-{{x}}^{{2}}+{4}{x}\right)}={\left(-{1}\right)}{\left({{x}}^{{2}}-{4}{x}\right)}{\quad\text{or}\quad}-{\left({{x}}^{{2}}-{4}{x}\right)}$

The $\displaystyle{{x}}^{{2}}-{4}{x}$ or $\displaystyle{{x}}^{{-{{4}}}}{x}+{0}$ resembles the $\displaystyle{a}{{x}}^{{2}}+{b}{x}+{c}$ where:

$\displaystyle{a}={1}$ , $\displaystyle{b}=-{4}$ and $\displaystyle{c}={0}$ .

To complete the square, we add and subtract $\displaystyle{{\left(-\frac{{b}}{{{2}{a}}}\right)}}^{{2}}$ .

Using $\displaystyle{a}={1}$ and $\displaystyle{b}=-{4}$ , we get:

$\displaystyle{{\left(-\frac{{b}}{{{2}{a}}}\right)}}^{{2}}={{\left(-\frac{{-{4}}}{{{2}{\left({1}\right)}}}\right)}}^{{2}}$

$\displaystyle={{\left(\frac{{4}}{{2}}\right)}}^{{2}}$

$\displaystyle={{2}}^{{2}}$

$\displaystyle={4}$

Add and subtract 4 inside the $\displaystyle{\left({{x}}^{{2}}-{4}{x}\right)}$ :

$\displaystyle-{\left({{x}}^{{2}}-{4}{x}+{4}-{4}\right)}$

Distribute the negative sign on -4 to rewrite it as:

$\displaystyle-{\left({{x}}^{{2}}-{4}{x}+{4}\right)}+{4}$

Factor the perfect square trinomial: $\displaystyle{{x}}^{{2}}-{4}{x}+{4}={{\left({x}-{2}\right)}}^{{2}}$ .

$\displaystyle-{{\left({x}-{2}\right)}}^{{2}}+{4}$

For the original problem, we let: $\displaystyle-{{x}}^{{2}}+{4}{x}=-{{\left({x}-{2}\right)}}^{{2}}+{4}$ :

$\displaystyle{2}\int\frac{{\left.{d}{x}}}{\sqrt{{-{{x}}^{{2}}+{4}{x}}}}={2}\int\frac{{\left.{d}{x}}}{\sqrt{{-{{\left({x}-{2}\right)}}^{{2}}+{4}}}}$

It can also be rewritten as:

$\displaystyle{2}\int\frac{{\left.{d}{x}}}{\sqrt{{-{{\left({x}-{2}\right)}}^{{2}}+{{2}}^{{2}}}}}={2}\int\frac{{\left.{d}{x}}}{\sqrt{{{{2}}^{{2}}-{{\left({x}-{2}\right)}}^{{2}}}}}$

The integral part resembles the integral formula:

$\displaystyle\int\frac{{{d}{u}}}{\sqrt{{{{a}}^{{2}}-{{u}}^{{2}}}}}={\arcsin{{\left(\frac{{u}}{{a}}\right)}}}+{C}$ .

Applying the formula, we get:

$\displaystyle{2}\int\frac{{\left.{d}{x}}}{\sqrt{{{{2}}^{{2}}-{{\left({x}-{2}\right)}}^{{2}}}}}={2}\cdot{\left(\frac{{\arcsin{{\left({x}-{2}\right)}}}}{{2}}\right)}+{C}$

Then the indefinite integral :

$\displaystyle\int\frac{{2}}{\sqrt{{-{{x}}^{{2}}+{4}{x}}}}{\left.{d}{x}\right.}={2}{\arcsin{{\left(\frac{{{x}-{2}}}{{2}}\right)}}}+{C}$

New Notes Blog

Sunday, 8 March 2015

$\displaystyle\int\frac{{2}}{\sqrt{{-{{x}}^{{2}}+{4}{x}}}}{\left.{d}{x}\right.}$ Find or evaluate the integral by completing the square

No comments:

Post a Comment

In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?

Sunday, 8 March 2015

Find or evaluate the integral by completing the square

No comments:

Post a Comment

In &quot;By the Waters of Babylon,&quot; under the leadership of John, what do you think the Hill People will do with their society?

$\displaystyle\int\frac{{2}}{\sqrt{{-{{x}}^{{2}}+{4}{x}}}}{\left.{d}{x}\right.}$ Find or evaluate the integral by completing the square

In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?