Question

How do you find the antiderivative of ${e^{2x}}dx$ ?

Answer 1

Hint : In mathematics, integration is the concept of calculus and it is the act of finding the integrals. Here we will find integration, by using the concept of equivalent value, where the same term will be multiplied and divided and will simplify and then will place the formula and simplify for the resultant answer. Here we will assume first for the part of integration, and then find its derivative and the function of the given integral with respect to “x” and then will place the value in the function and then simplify it for the resultant value. We will take the given function equal to I (integration)

Complete step-by-step answer :
$I = \int {{e^{2x}}} dx$
Find the equivalent function of the above equation. Multiply and divide the above equation with the number $2$
$I = \dfrac{1}{2}\int {2{e^{2x}}} dx$
We can see that $2dx = d(2x)$ that means $2$ is the derivative of $2x$ .
We can follow this as –
$I = \dfrac{1}{2}\int {{e^{2x}}} d(2x)$
Let us assume that $u = 2x$ and place it in the above equation –
$I = \dfrac{1}{2}\int {{e^u}} d(u)$
Apply the identity- $I = \int {{e^u}} d(u) = {e^u}$ in the above equation –
$I = \dfrac{1}{2}{e^u}$
Replace the value $u = 2x$ in the above equation
$I = \dfrac{1}{2}{e^{2x}}$
This is the required equation –
$\int {{e^{2x}}} dx = \dfrac{1}{2}{e^{2x}}$
So, the correct answer is “ $\dfrac{1}{2}{e^{2x}}$ ”.

Note : Anti-derivative is the another name of the inverse derivative, the primitive function and the primitive integral or the indefinite integral of a function f is the differentiable function F whose derivative is equal to the original function f. Know the difference between the differentiation and the integration and apply formula accordingly. Differentiation can be represented as the rate of change of the function, whereas integration represents the sum of the function over the range. They are inverses of each other.