# How to find the integral of a derivative?

## Question:

How to find the integral of a derivative?

## IAntiderivative:

Integration and differentiation are closely related. Indefinite integration is a process of finding the antiderivative of the given function, basically it is opposite of derivative.

Indefinite integration of a function {eq}f(x) {/eq} can be written as {eq}\displaystyle \int f(x) \, \mathrm{d}x . {/eq} Where dx is the differential of x.

We can find out the general form of the function by integrating the first order derivative. If {eq}F(x) {/eq} is the antiderivative of {eq}f(x) {/eq}. Then {eq}\displaystyle \int f(x)\ dx = F(x)+C {/eq}

Let us consider a separable first order differential equation {eq}\displaystyle \frac{dy}{dx} = f(x) {/eq}

To compute the integration first we'll rewrite the differential equation in separable form:

{eq}\Rightarrow \displaystyle \ dy = f(x)\ dx {/eq}

Integrate both sides:

{eq}\Rightarrow \displaystyle\int \ dy =\int f(x)\ dx {/eq}

If {eq}F(x) {/eq} is the antiderivative of {eq}f(x) {/eq}. Then we can write {eq}\displaystyle \int f(x)\ dx = F(x)+C {/eq}

{eq}\Rightarrow \displaystyle y = F(x)+C {/eq} ( where C is an arbitrary constant.)

Hence, the solution is {eq}\displaystyle y(x) = F(x)+C. {/eq}

Therefore, we can compute the integral of a derivative to find out the general form of the function.