`int dx / sqrt(1-(x+1)^2)` Find the indefinite integral

Thursday, March 31, 2011

$\displaystyle\int\frac{{\left.{d}{x}}}{\sqrt{{{1}-{{\left({x}+{1}\right)}}^{{2}}}}}$ Find the indefinite integral

Indefinite integral are written in the form of $\displaystyle\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={F}{\left({x}\right)}+{C}$

where: f(x) as the integrand

F(x) as the anti-derivative function

C as the arbitrary constant known as constant of integration

For the given problem, the integrand $\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{\sqrt{{{1}-{{\left({x}+{1}\right)}}^{{2}}}}}$ we apply

u-substitution by letting $\displaystyle{u}={\left({x}+{1}\right)}$ and $\displaystyle{d}{u}={1}{\left.{d}{x}\right.}{\quad\text{or}\quad}{d}{u}={\left.{d}{x}\right.}$ .

$\displaystyle\int\frac{{{\left.{d}{x}\right.}}}{\sqrt{{{1}-{{\left({x}+{1}\right)}}^{{2}}}}}=\int\frac{{{d}{u}}}{\sqrt{{{1}-{{u}}^{{2}}}}}$

$\displaystyle\int\frac{{{d}{u}}}{\sqrt{{{1}-{{u}}^{{2}}}}}$ resembles the basic integration formula for inverse sine function: $\displaystyle\int\frac{{{\left.{d}{x}\right.}}}{\sqrt{{{1}-{{x}}^{{2}}}}}={\arcsin{{\left({x}\right)}}}+{C}$ .

By applying the formula, we get:

$\displaystyle\int\frac{{{d}{u}}}{\sqrt{{{1}-{{u}}^{{2}}}}}={\arcsin{{\left({u}\right)}}}+{C}$

Then to express it in terms of x, we substitute $\displaystyle{u}={\left({x}+{1}\right)}$ :

$\displaystyle{\arcsin{{\left({u}\right)}}}+{C}={\arcsin{{\left({x}+{1}\right)}}}+{C}$

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Thursday, March 31, 2011

$\displaystyle\int\frac{{\left.{d}{x}}}{\sqrt{{{1}-{{\left({x}+{1}\right)}}^{{2}}}}}$ Find the indefinite integral

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Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Thursday, March 31, 2011

Find the indefinite integral

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Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

$\displaystyle\int\frac{{\left.{d}{x}}}{\sqrt{{{1}-{{\left({x}+{1}\right)}}^{{2}}}}}$ Find the indefinite integral

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?