Basic Integration Rules

Antiderivatives

In many cases in calculus, it is known that the rate of change of one variable with respect to another, but we do not have a formula which relates the variables. In other words, it is known that $\dfrac{dy}{dx}$, but we need to know $y$ in terms of $x$.
The process of finding $y$ from $\dfrac{dy}{dx}$, or $f(x)$ from $f'(x)$, is the reverse process of differenciation. We call it antidifferentiation or integration.
$$ y\xrightarrow{\text{differentiation}} \dfrac{dy}{dx}$$ $$ y\xleftarrow[\text{or integration}]{\text{antidifferentiation}} \dfrac{dy}{dx}$$ $$ f(x)\xrightarrow{\text{differentiation}} f'(x)$$ $$ f(x)\xleftarrow[\text{or integration}]{\text{antidifferentiation}} f'(x)$$ Consider $\dfrac{dy}{dx}=x^3$.
From our work on differentiation, we know that when we differentiate power functins the index reduces by $1$. We hence know that $y$ must involve $x^4$.
If $y=x^4$ then $\dfrac{dy}{dx}=4x^3$, so if we start with $y=\dfrac{1}{4}x^4$ then $\dfrac{dy}{dx}=x^4$.
However, in all of the cases $y=\dfrac{1}{4}x^4+1$, $y=\dfrac{1}{4}x^4+-2$, $y=\dfrac{1}{4}x^4+100$ and $y=\dfrac{1}{4}x^4+12$ we find $\dfrac{dy}{dx}=x^3$.
In fact, there are infinitely many functions of the form $y=\dfrac{1}{4}x^4+c$ where $c$ is an arbitrary constant which will give $\dfrac{dy}{dx}=x^3$.
Ignoring the arbitrary constant, we say that $\dfrac{1}{4}x^4$ is the antiderivative of $x^3$. It is the simplest function which, when differentiated, gices $x^2$.

Example 1

Find the antiderivative of $x^5$.

Example 2

Find the antiderivative of $\sqrt{x}$.

Example 3

Find the antiderivative of $(2x+1)^3$.

Integration

\( \begin{align} \displaystyle x^3 &= \dfrac{d}{dx}\Big(\dfrac{1}{4}x^4\Big) \\ x^3 &= \dfrac{d}{dx}\Big(\dfrac{1}{4}x^4 +1\Big) \\ x^3 &= \dfrac{d}{dx}\Big(\dfrac{1}{4}x^4 +10\Big) \\ x^3 &= \dfrac{d}{dx}\Big(\dfrac{1}{4}x^4 -7\Big) \\ \therefore \int{x^3}dx &= \dfrac{1}{4}x^4+c \\ \end{align} \)

Example 4

If $y=x^4+3x^3$, find $\dfrac{dy}{dx}$. Hence find $\displaystyle \int{(4x^3+9x^2)}dx$.

Rules for Integration

$$\displaystyle \int{x^n}dx = \dfrac{1}{n+1}x^{n+1}+c$$

Example 5

Find $\displaystyle \int{x^6}dx$.

Example 6

Find $\displaystyle \int{\sqrt{x}}dx$.

Example 7

Find $\displaystyle \int{5}dx$.

Example 8

Find $\displaystyle \int{(3x^3 + 4x^4)}dx$.