Question

How do you integrate \[{e^{4x}}dx\]?

Answer 1

Hint: Here we will use the basic concept of the integration. We will integrate the given exponential function using the standard formula for the integration of an exponential to get the integration of the required equation. Integration is defined as the summation of all the discrete data.

Complete step by step solution:
Given equation is \[{e^{4x}}dx\].
Here, we have to integrate an exponential function with respect to $x$.
Let \[I\] be the integration of the given equation i.e \[{e^{4x}}dx\]. Therefore we can write the integration as
\[I = \int {{e^{4x}}dx} \]
Now we will use the standard formula of the integration of the exponential function. The standard formula is\[\int {{e^{ax}}dx} = \dfrac{1}{a}{e^{ax}} + C\].
So, we will apply this formula for the integration of the given equation.
Now the value of the constant \[a\] will become 4 i.e. \[a = 4\]. Therefore by integrating and putting the value of constant we get
\[ \Rightarrow I = \int {{e^{4x}}dx} = \dfrac{1}{4}{e^{4x}} + C\]

Hence the integration of the given equation i.e. \[{e^{4x}}dx\] is equal to \[\dfrac{1}{4}{e^{4x}} + C\].

Note:
Here, we need to remember that we have to put the constant term \[C\] after the integration of an equation and the value of the constant can be anything i.e. it can be zero or any value. Differentiation is the inverse function of the integration i.e. differentiation of the integration of a function is equal to the value of the function or integration of the differentiation of a function is equal to the value of the function.
We should also know the basic formula of the integration by parts of an equation.
Basic formula of the integration of \[u \times v\] is \[\int {\left( {u \times v} \right)dx} = u\int {vdx} - \int {\left( {u'\int {vdx} } \right)dx} \].