Question

What is the antiderivative of ${{3}^{x}}dx$?

Answer 1

To obtain the antiderivative of ${{3}^{x}}dx$ we will use relation between logarithm and exponential function. Firstly using log and exponential function relation we will get our function in such a term which is easily integrated where $y={{e}^{\ln \left( y \right)}}$. Next we will use the formula for integrating logarithm function and simplify it further to get our desired answer.

Complete step-by-step solution:
To find antiderivative of ${{3}^{x}}dx$ let us make it in form of logarithm and exponential function as:
$y={{e}^{\ln \left( y \right)}}$
So by using above concept we get,
${{3}^{x}}={{e}^{\ln \left( {{3}^{x}} \right)}}$
As we know the property of logarithm i.e. $\log {{a}^{^{b}}}=b\log a$ applying above we get,
${{3}^{x}}={{e}^{x\ln \left( 3 \right)}}$……..$\left( 1 \right)$
On integrating equation (1) both side with respect to $x$ we get,
$\int{{{3}^{x}}dx=\int{{{e}^{x\ln \left( 3 \right)}}dx}}$
Using the formula of exponential given below in above equation:
$\int{{{e}^{\beta x}}dx}=\dfrac{1}{\beta }{{e}^{\beta x}}+C$
Where, $C$ is any constant.
So on using the formula we get,
$\int{{{3}^{x}}dx=\dfrac{1}{\ln \left( 3 \right)}{{e}^{x\ln \left( 3 \right)}}}+C$
Put the value from equation (1) in above equation for simplifying it further we get,
$\int{{{3}^{x}}dx=\dfrac{1}{\ln \left( 3 \right)}{{3}^{x}}}+C$
Hence, anti-derivative of ${{3}^{x}}dx$ is $\dfrac{{{3}^{x}}}{\ln \left( x \right)}+C$ where $C$ is any constant.

Note: The anti-derivative of a function is done to go backward from the derivative of the function to the function itself. As the derivatives don’t determine the function completely we add a constant with the solution. Anti-derivative is commonly known as indefinite integral. That is why a derivative can have many antiderivatives. A simple definition that define the relation between derivative and antiderivative is that a function $F$ is an anti-derivative of the function $f$ if ${F}'\left( x \right)=f\left( x \right)$ for all the $x$ in the domain of $f$. The symbol used for anti-derivative is $\int{{}}$ and $\int{f\left( x \right)dx}$ is known as indefinite integral of $f$