Question

How do you integrate $\int {{e^{5x}}(5)dx} ?$

Answer 1

We have been given a role in the given question. The function is a function of the Euler number. As an argument, this function contains a variable. This role is then elevated to strength. We have to calculate the integral value of this whole function. We will need to find the primitive function to overcome the integral, which gives the expression in the question when differentiated. Separate the constant part or take it out of the integration.

Complete step by step solution:
In this question, to evaluate the given integral $\int {{e^{5x}}(5)dx}$ we will use the formula for derivation of Euler’s number function which is given as follows
$\dfrac{{d{e^{ax}}}}{{dx}} = a{e^{ax}}$
Now we know that integration and differentiation are inverse functions of each other, so we have to find the expression whose differentiation will be the given integrand.
On observing clearly, we can see that,
$\dfrac{{d\left( {{e^{5x}} + c} \right)}}{{dx}} = 5{e^{5x}},\;{\text{where}}\;c$ is an arbitrary constant

Therefore we can say that $\int {{e^{5x}}(5)dx} = {e^{5x}} + c$

Note: The given integral can be integrated directly either with the use of $uv$ integration or first separating the constant part in the integrand from the integral and then integrating the rest of the function using integration formula for Euler number.