`int sec^2(3x) dx` Find the indefinite integral

Saturday, January 14, 2012

$\displaystyle\int{{\sec}}^{{2}}{\left({3}{x}\right)}{\left.{d}{x}\right.}$ 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: $\displaystyle{f{{\left({x}\right)}}}$ as the integrand

$\displaystyle{F}{\left({x}\right)}$ as the anti-derivative function

$\displaystyle{C}$ as the arbitrary constant known as constant of integration

For the given problem: $\displaystyle\int{{\sec}}^{{2}}{\left({3}{x}\right)}{\left.{d}{x}\right.}$ , the integrand function: $\displaystyle{f{{\left({x}\right)}}}={{\sec}}^{{2}}{\left({3}{x}\right)}$ is in a form of trigonometric function.

To solve for the indefinite integral, we may apply the basic integration formula for secant function:

$\displaystyle\int{{\sec}}^{{2}}{\left({u}\right)}{d}{u}={\tan{{\left({u}\right)}}}+{C}$

We may apply u-substitution when by letting:

$\displaystyle{u}={3}{x}$ then $\displaystyle{d}{u}={3}{\left.{d}{x}\right.}$ or $\displaystyle\frac{{{d}{u}}}{{3}}={\left.{d}{x}\right.}$ .

Plug-in the values of $\displaystyle{u}={3}{x}$ and $\displaystyle{\left.{d}{x}\right.}=\frac{{{d}{u}}}{{3}}$ , we get:

$\displaystyle\int{{\sec}}^{{2}}{\left({3}{x}\right)}{\left.{d}{x}\right.}=\int{{\sec}}^{{2}}{\left({u}\right)}\cdot\frac{{{d}{u}}}{{3}}$

$\displaystyle=\int\frac{{{{\sec}}^{{2}}{\left({u}\right)}}}{{3}}{d}{u}$

Apply basic integration property: $\displaystyle\int{c}\cdot{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}$ .

$\displaystyle\int\frac{{{{\sec}}^{{2}}{\left({u}\right)}}}{{3}}{d}{u}={\left(\frac{{1}}{{3}}\right)}\int{{\sec}}^{{2}}{\left({u}\right)}{d}{u}$

Then following the integral formula for secant, we get:

$\displaystyle{\left(\frac{{1}}{{3}}\right)}\int{{\sec}}^{{2}}{\left({u}\right)}{d}{u}=\frac{{1}}{{3}}{\tan{{\left({u}\right)}}}+{C}$

Plug-in $\displaystyle{u}={3}{x}$ to solve for the indefinite integral F(x):

$\displaystyle\int{{\sec}}^{{2}}{\left({3}{x}\right)}{\left.{d}{x}\right.}=\frac{{1}}{{3}}{\tan{{\left({3}{x}\right)}}}+{C}$

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Saturday, January 14, 2012

$\displaystyle\int{{\sec}}^{{2}}{\left({3}{x}\right)}{\left.{d}{x}\right.}$ 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?

Saturday, January 14, 2012

Find the indefinite integral

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Post a Comment

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

$\displaystyle\int{{\sec}}^{{2}}{\left({3}{x}\right)}{\left.{d}{x}\right.}$ 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?