### Integration: Antiderivatives

### The antiderivative of a function

Antiderivative

The function #\orange {F}# is an **antiderivative** of the function #\blue f# if \[\orange F'(x)=\blue f(x)\]

We denote the antiderivative of #\blue f# as follows:

\[\begin{array}{rcl}\displaystyle \int \blue {f(x)} \; \dd x \end{array}\]

This is also called the **indefinite integral**.

The result of the indefinite integral are functions of the form #\orange F(x) + \green C # where #\orange F# is an antiderivative of #\blue f# and #\green C# is a constant, because the constant is eliminated when differentiating.

We call #\green C# the **constant** **of** **integration**.

\[\begin{array}{rcl}\blue f(x)&=&3x^2 \\ \text{gives i.e.} \\ \orange F(x)&=&x^3 \\ \orange F(x)&=&x^3 + \green{3} \\ \orange F(x) &=&x^3 + \green{5} \\ \\ \text{hence} \\\displaystyle \int \blue {3x^2} \; \dd x &=& x^3+\green{C} \\ \text{because} \\ \dfrac{\dd}{\dd x} (x^3+\green C) &=& 3x^2\end{array}\]

\[

f(x)=4\cdot x^3+7

\]

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