Question

integrate: $e^{2x}$

Answer 1

The final answer is $\boxed{\frac{1}{2}e^{2x} + C}$

Answer 2

To integrate the function $e^{2x}$, we can use a simple substitution method. Let $u = 2x$. Differentiating both sides with respect to $x$, we get: $\frac{du}{dx} = 2$ This implies $dx = \frac{1}{2} du$.

Now, substitute $u$ and $dx$ into the integral: $\int e^{2x} dx = \int e^u \left(\frac{1}{2} du\right)$ We can pull the constant $\frac{1}{2}$ out of the integral: $\frac{1}{2} \int e^u du$ The integral of $e^u$ with respect to $u$ is $e^u$. So, the result is: $\frac{1}{2} e^u + C$ Finally, substitute back $u = 2x$: $\frac{1}{2} e^{2x} + C$