Question

How do you integrate $ \int {e^x}{e^x} $ using substitution?

Answer 1

Hint: In order to this question, to integrate the given expression by substitution by following the formula $ {a^b}({a^c}) = {a^{b + c}} $ and then we will do further substitution for the given expression.

Complete step by step solution:
We will integrate the given expression by using the rule $ {a^b}({a^c}) = {a^{b + c}} $ to rewrite the integral as-
 $ \because \int {e^x}{e^x}dx = \int {e^{2x}}dx $
Now substitute $ u = 2x $
so, we do differentiation of the upper assumed equation:
 $ \begin{gathered}
   \Rightarrow \dfrac{{du}}{{dx}} = 2 \\
   \Rightarrow du = 2.dx \\
\end{gathered} $
Since, $ \int {e^u}du = {e^u} $ :
 $ \dfrac{1}{2}\int {e^u}du = \dfrac{1}{2}{e^u} = \dfrac{{{e^u}}}{2} = \dfrac{{{e^{2x}}}}{2} + C $
So, the correct answer is “ $ \dfrac{{{e^{2x}}}}{2} + C $ ”.

Note: In calculus, integration by substitution, also known as u-substitution or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards".