Question

Integrate the following: e^(2x)

Answer 1

You can integrate it by substitution:

$\large\begin{array}{l} \mathsf{\displaystyle\int\!e^{2x}\,dx}\\\\ =\mathsf{\displaystyle\int\!\frac{1}{2}\cdot 2e^{2x}\,dx}\\\\ =\mathsf{\displaystyle\frac{1}{2}\int\!e^{2x}\cdot 2\,dx\qquad(i)} \end{array}$


$\large\begin{array}{l} \textsf{Now substitute}\\\\ \mathsf{2x=u\quad\Rightarrow\quad 2\,dx=du}\\\\\\ \textsf{then (i) becomes}\\\\ =\mathsf{\displaystyle\frac{1}{2}\int\!e^u\,du}\\\\ =\mathsf{\dfrac{1}{2}\cdot e^u+C}\\\\ =\mathsf{\dfrac{1}{2}\cdot e^{2x}+C} \end{array}$


$\large\begin{array}{l} \boxed{\begin{array}{l} \mathsf{\displaystyle\int\!e^{2x}\,dx=\frac{1}{2}\cdot e^{2x}+C} \end{array}}\qquad\checkmark \end{array}$


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$\large\textsf{I hope it helps. :-)}$


Tags: integrate indefinite integral substitution exponential composite integral calculus