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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}

Answer and Explanation:

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.


Learn more about this topic:

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Antiderivative: Rules, Formula & Examples

from Calculus: Help and Review

Chapter 8 / Lesson 12
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