Integral Toolbox 2 Integral Toolbox 2
Introduction
We will look at each derivative rule and reverse it for antiderivatives.
The Power Rule
The Constant Rule
The Sum Rule
The Product Rule
The Quotient Rule
The Chain Rule
The Constant Product Rule
The Power Rule
\begin{align*}
\frac{d}{dx} x^n &= nx^{n-1} \\\\
-&-- \\\\
\int nx^{n-1} \,dx &= x^n + C \\\\
\int 3x^2 \,dx &= x^3 + C \\\\
\end{align*}
The Constant Rule
\begin{align*}
\frac{d}{dx} C &= 0 \\\\
-&-- \\\\
\int 0 \,dx &= C
\end{align*}
The Sum Rule
\begin{align*}
\frac{d}{dx} [f(x) + g(x)] &= \frac{d}{dx} f(x) + \frac{d}{dx} g(x) \\\\
-&-- \\\\
\int [f(x) + g(x)] \,dx &= \int f(x) \,dx + \int g(x) \,dx \\\\
\int [3x^2 + 6x^5] \,dx &= \int 3x^2 \,dx + \int 6x^5 \,dx \\\\
\int [3x^2 + 6x^5] \,dx &= x^3 + x^6 + C
\end{align*}
The Product Rule
\begin{align*}
\frac{d}{dx} [f(x)g(x)] &= f(x) \frac{d}{dx} g(x) + g(x) \frac{d}{dx} f(x) \\\\
-&-- \\\\
\int [f(x)g'(x) + g(x)f'(x)] \,dx &= f(x)g(x) + C \\\\
\int [x^3 \cdot 6x^5 + x^6 \cdot 3x^2] \,dx &= x^3 \cdot x^6 + C \\\\
\int \cos(2x) \, dx &= \int [\cos^2(x) - \sin^2(x)] \, dx \\\\
\int \cos(2x) \, dx &= \sin(x)\cos(x) + C
\end{align*}
The Quotient Rule
\begin{align*}
\frac{d}{dx} [\frac{f(x)}{g(x)}] &= \frac{g(x) \frac{d}{dx} f(x) - f(x) \frac{d}{dx} g(x)}{g(x)^2} \\\\
-&-- \\\\
\int [\frac{g(x)f'(x) - f(x)g'(x)}{g(x)^2}] \,dx &= \frac{f(x)}{g(x)} + C \\\\
\int [\frac{x^6 \cdot 3x^2 - x^3 \cdot 6x^5}{x^{12}}] \,dx &= \frac{x^3}{x^6} + C = x^{-3} + C
\end{align*}
The Chain Rule
\begin{align*}
\frac{d}{dx} f(g(x)) &= f'(g(x)) \cdot g'(x) \\\\
-&-- \\\\
\int f'(g(x)) \cdot g'(x) \,dx &= f(g(x)) + C \\\\
\int 2\cos(2x) \,dx &= \sin(2x) + C
\end{align*}
The Constant Product Rule
\begin{align*}
\frac{d}{dx} [a \cdot f(x)] &= a \cdot \frac{d}{dx} f(x) \\\\
-&-- \\\\
\int a \cdot f'(x) \,dx &= a \cdot f(x) + C \\\\
\int 6x^2 \,dx &= 2 \int 3x^2 \,dx = 2x^3 + C \\\\
\int \cos(2x) \,dx &= \frac{1}{2} \int 2\cos(2x) \,dx = \frac{1}{2} \sin(2x) + C
\end{align*}