Question

Find the anti - derivative (or integral) of the given function by the method of inspection.
${e^{2x}}$.

Answer 1

In this question, an anti – derivative is the opposite of derivatives. An anti – derivative is a function that reverses what the derivative does. One function has many anti – derivatives, but they all take the form of a function plus an arbitrary constant. Anti – derivative is the key part of indefinite integrals. The anti – derivative is the operation that goes backward from the derivative of a function to the function itself.

Complete step-by-step answer:
Here we have;
Given function is ${e^{2x}}$
Differentiating ${e^{2x}}$ with respect to x using chain rule, we get
$\dfrac{d}{{dx}}\left( {{e^{2x}}} \right) = 2{e^{2x}}$
Rearranging terms
$\Rightarrow {e^{2x}} = \dfrac{1}{2}\dfrac{d}{{dx}}\left( {{e^{2x}}} \right)$
$\therefore {e^{2x}} = \dfrac{d}{{dx}}\left( {\dfrac{1}{2}{e^{2x}}} \right)$
Anti-derivative or integral of ${e^{2x}}$ $= \dfrac{1}{2}{e^{2x}}$.

Note: An anti - derivative of a function f is a function whose derivative is f. To find an antiderivative for a function f, we can often reverse the process of differentiation. The process of solving for anti - derivatives is called anti - differentiation (or indefinite integration), and its opposite operation is called differentiation, which is the process of finding a derivative. Anti – derivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an anti – derivative evaluated at the endpoints of the interval. If F(x) is a function with F’(x) = f(x), then we say that F(x) is an anti – derivative of f(x). Every continuous function has an anti - derivative, and in fact has infinitely many anti – derivatives. Two anti - derivatives for the same function f(x) differ by a constant.