`int 3^xdx` Find the indefinite integral

Tuesday, October 12, 2010

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 $\displaystyle\int{{3}}^{{x}}{\left.{d}{x}\right.}$ has a integrand in a form of exponential function.

There is basic integration formula for exponential function:

$\displaystyle\int{{a}}^{{u}}{d}{u}=\frac{{{a}}^{{u}}}{{{\ln{{\left({a}\right)}}}}}+{C}$ where a is a constant.

By comparison, $\displaystyle{a}={3}$ , $\displaystyle{u}={x}$ , and $\displaystyle{d}{u}={\left.{d}{x}\right.}$ .

Applying the formula, we get:

$\displaystyle\int{{3}}^{{x}}{\left.{d}{x}\right.}=\frac{{{3}}^{{x}}}{{{\ln{{\left({3}\right)}}}}}+{C}$

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