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".