Question

How do you evaluate the integral of \[\int {{e^{\left( {5x} \right)}}dx} \]?

Answer 1

Hint: In the given question, we have been given a function. The function is an Euler’s number function. This function contains a variable as an argument. Then this function is raised to a power. We have to calculate the value of the integral of this whole function. To solve the integral, we are going to need to find the primitive function which when differentiated gives the expression in the question.

Formula Used:
In the question, we are going to use the formula of derivation of Euler’s number function:
\[\dfrac{{d\left( {{e^x}} \right)}}{{dx}} = \left( {{e^x}} \right) \times \dfrac{{dx}}{{dx}} = {e^x}\]

Complete step by step answer:
The given expression to be integrated is \[\int {{e^{\left( {5x} \right)}}dx} \].
To solve the integral, we need to find the primitive function which when differentiated gives the expression in the question. And clearly, we can get that,
\[\dfrac{{d\left( {\dfrac{{{e^{5x}}}}{5} + c} \right)}}{{dx}} = {e^{5x}}\]

Hence, \[\int {{e^{\left( {5x} \right)}}dx} = \dfrac{{{e^{5x}}}}{5} + c\]

Note:
In the given question, we had to find the value of the integral of an Euler’s number function. There can be many ways by which we can attempt to find the answer, but we use the reverse thinking way as it is the easiest way. We figure that out by seeing the \[dx\] part – the differential is of the argument of the determining function – the Euler’s number function. So, we must know the concept of the things being used.