Question

What is the integral of \[{{e}^{7x}}\]?

Answer 1

Hint: From the question given, we have been asked to find the integral of \[{{e}^{7x}}\]. To solve this question, we have to know the basic concepts of integration. We have to use the substitution method to solve the given question. So, we will take u=$7x$ and then proceed.

Complete step by step answer:
Let’s learn what integration is before understanding the concept of integration by substitution. The integration of a function \[f\left( x \right)\] is given by \[F\left( x \right)\] and it is represented by:
\[\int{f\left( x \right)}dx\] = \[F\left( x \right)\] + C
Substitution method: In this method of integration by substitution, any given integral is transformed into a simple form of integral by substituting the independent variable by others. This method is also known as u- substitution or change of variables, is a method for evaluating integrals and antiderivatives.
Here in this case, \[{{e}^{7x}}\]
We are going to use u- substitution.
Let u = \[7x\]
Differentiate(derivative) both parts:
\[du=7dx\]
\[\dfrac{du}{7}=dx\]
Now we can replace everything in the integral:
\[\int{\dfrac{1}{7}}{{e}^{u}}du\]
Bring the constant upfront
\[\dfrac{1}{7}\int{{{e}^{u}}}du\]
The integral of \[{{e}^{u}}\] is simply \[{{e}^{u}}\]
\[\dfrac{1}{7}{{e}^{u}}\]
And replace the u back
\[\dfrac{1}{7}{{e}^{7x}}\]
There is also a shortcut you can use:
Whenever you have a function of which you know the integral\[f\left( x \right)\], but it has a different argument
\[\Rightarrow \] the function is in the form \[f\left( ax\pm b \right)\]
If you want to integrate this, it is always equal to \[\dfrac{1}{a}F\left( ax+b \right)\], where \[F\] is the integral of the regular \[f\left( x \right)\] function.
In this case:
\[f\left( x \right)={{e}^{x}}\]
\[F\left( x \right)=\int{{{e}^{x}}}dx={{e}^{x}}\]
\[\begin{align}
  & \\
 & a=7 \\
 & b=0 \\
 & f\left( ax+b \right)={{e}^{7x}} \\
\end{align}\]
\[\Rightarrow \int{{{e}^{7x}}}dx=\dfrac{1}{a}F\left( ax+b \right)=\dfrac{1}{7}{{e}^{7x}}\]

Note: Students should be well known about the concept of integration. Students should know the formulas in integration. Students should know the method of substitution. Students should be careful while performing substitution methods. Students should be careful while calculating the problem.