Question

How do you integrate ${{3}^{x}}$ ?

Answer 1

When the base is constant and its power is variable, the we use the formula of integration of this type of function, ${{\int{a}}^{x}}=\dfrac{{{a}^{x}}}{\ln a}$ that means the integration is just the same function but divided by logarithm of $a$ with base $e$ .

Complete step by step solution:
Since we have to find the integration of ${{3}^{x}}$
Let’s assume the integration of given function be $z$
$\Rightarrow z=\int{{{3}^{x}}}dx$
Now we know that the integration of ${{a}^{x}}$ is $\dfrac{{{a}^{x}}}{\ln a}$
$\Rightarrow {{\int{a}}^{x}}=\dfrac{{{a}^{x}}}{\ln a}$
On comparing the given question with the above formula
$\Rightarrow a=3$
$\Rightarrow \int{{{3}^{x}}}=\dfrac{{{3}^{x}}}{\ln 3}$
The above term can be written as
$(\because \ln 3={{\log }_{e}}3)$
$\Rightarrow \int{{{3}^{x}}}=\dfrac{{{3}^{x}}}{{{\log }_{e}}3}$

Hence the integration of ${{3}^{x}}$ is $\dfrac{{{3}^{x}}}{{{\log }_{e}}3}$

Note:
During integrating the base is constant and the exponential power is variable, it will confuse as generally the base is variable and the power is constant. When the base is constant and power is variable the integration, as well as differentiation, is in terms of a logarithm.