Standard Results
Fundamental Theorem of Calculus
The process of differentiation is the inverse of that of integration.
\[ \ddx{}{x} \int{f(x)\ \dx{x}} = f(x) \]
Standard Results
|
Derivative \( \ddx{}{x} f(x) \) |
Function \( f(x) \) |
Integral \( \int{f(x)\ \dx{x}} \) |
|
|---|---|---|---|
| Polynomial | \[ 0 \] | \[ c \] | \[ cx \] |
| \[ nx^{n-1} \] | \[ x^n \] | \[ \frac{1}{n+1} x^{n+1} \] | |
| Exponential | \[ ae^{ax} \] | \[ e^{ax} \] | \[ \frac{1}{a} e^{ax} \] |
| \[ \frac{1}{x} \] | \[ \ln{x} \] | \[ x \ln{x} + x \] | |
| Trigonometric | \[ \cos{x} \] | \[ \sin{x} \] | \[-\cos{x} \] |
| \[-\sin{x} \] | \[ \cos{x} \] | \[ \sin{x} \] | |
| \[ \sec^2{x} \] | \[ \tan{x} \] | \[ -\ln(\cos{x}) \] | |
| \[ -\csc^2{x} \] | \[ \cot{x} \] | \[ \ln(\sin{x}) \] | |
| \[ \sec{x} \tan{x} \] | \[ \sec{x} \] | \[ \ln\left( \sin\frac{x}{2} + \cos\frac{x}{2} \right) - \ln\left( \cos\frac{x}{2} - \sin\frac{x}{2} \right) \] | |
| \[ -\csc{x}\cot{x} \] | \[ \csc{x} \] | \[ \ln\left( \sin\frac{x}{2} \right) - \ln\left( \cos\frac{x}{2} \right) \] | |
| \[ \frac{1}{\sqrt{a^2 - x^2}} \] | \[ \frac{1}{a} \sin^{-1}\left( \frac{x}{a} \right) \] | ||
| \[-\frac{1}{\sqrt{a^2 - x^2}} \] | \[ \frac{1}{a} \cos^{-1}\left( \frac{x}{a} \right) \] | ||
| \[ \frac{1}{a^2 + x^2} \] | \[ \frac{1}{a} \tan^{-1}\left( \frac{x}{a} \right) \] | ||
| Hyperbolic | \[ \cosh{x} \] | \[ \sinh{x} \] | \[ \cosh{x} \] |
| \[ \sinh{x} \] | \[ \cosh{x} \] | \[ \sinh{x} \] | |
| \[ \sec^2{x} \] | \[ \tanh{x} \] | \[ \ln(\cosh{x}) \] | |
| \[ \frac{1}{\sqrt{1 + x^2}} \] | \[ \sinh^{-1}{x} \] | ||
| \[ \frac{1}{\sqrt{x^2 - 1}} \] | \[ \cosh^{-1}{x} \] | ||
| \[ \frac{1}{1 - x^2} \] | \[ \tanh^{-1}{x} \] |
Differentiation Laws
Constant Rule
\[ \ddx{}{x} (c u) = c \ddx{u}{x} \]
Product Rule
\[ \ddx{}{x} (u v) = v\ddx{u}{x} + u\ddx{v}{x} \]
Quotient Rule
\[ \ddx{}{x} \left( \frac{u}{v} \right) = \frac{ v\ddx{u}{x} - u\ddx{v}{x} }{v^2} \]
Chain Rule
\[ \ddx{u}{x} = \ddx{u}{y} \ddx{y}{x} \]
Implicit Differentiation
Implicit differentiation is used to differentiate functions such as f(y) in terms of another variable using the chain rule:
\[ \ddx{}{x}f(y) = \ddx{}{y}f(y) \times \ddx{y}{x} \]
Parametric Differentiation
Parametric differentiation is used to differentiate parametrically defined functions using the chain rule
Given that \( y=f(x), x=g(t), y=h(t) \):
\[ \ddx{y}{x} = \ddx{y}{t} \Big/ \ddx{x}{t} = \ddx{y}{t} \ddx{t}{x} \]
\[ \dddx{y}{x} = \ddx{}{x}\ddx{y}{x} = \left(\ddx{}{t}\ddx{y}{x}\right) \Big/ \ddx{x}{t} \]
Logarithmic differentiation
Logarithmic differentiation is used to differentiate complicated functions such as functions raised to powers
\[ \begin{align} y &= f(x) \\ \ln y &= \ln f(x) \\ \frac{1}{y}\ddx{y}{x} &= \ddx{}{x} \ln f(x) \end{align} \]
Integration Laws
Integration by Simple Substitution
Given \( u = f(x) , \ddx{u}{x} = f'(x) \therefore \dx{x} = \frac{\dx{u}}{f'(x)} \):
\[ \int_a^b g \left( f(x) \right) \dx{x} = \int_{f(a)}^{f(b)} g(u) \frac{\dx{u}}{f'(x)} \]
Integration by Parts
Integrate the product rule with respect to \( x \):
\[ \int{u\ddx{v}{x}\dx{x}} = uv - \int{v\ddx{u}{x}\dx{x}} \]
Integration by Substitution
Integrate the chain rule with respect to \( x \):
\[ \int{f'(g(x))g'(x)\ \dx{x}} = f(g(x)) + c \]
Given \( t = g(x) \):
\[ \int{f'(t) \ddx{t}{x} \dx{x}} = \int{ f'(t)\ \dx{t}} = f(t) + c = f(g(x)) + c \]
Common Substitutions
| Function | Substitution | Substituted Derivative |
|---|---|---|
| \[ \sqrt{a^2 - x^2} \] | \[ x = a \sin{\theta} \] | \[ \ddx{x}{\theta} = a \cos{\theta} \] |
| \[ x = a \tanh{u} \] | \[ \ddx{x}{u} = a \sech^2{u} \] | |
| \[ \sqrt{a^2 + x^2} \] | \[ x = a \sinh{u} \] | \[ \ddx{x}{u} = a \cosh{u} \] |
| \[ x = a \tan{\theta} \] | \[ \ddx{x}{\theta} = a \sec^2{\theta} \] | |
| \[ \sqrt{x^2 - a^2} \] | \[ x = a \cosh{u} \] | \[ \ddx{x}{u} = a \sinh{u} \] |
| \[ x = a \sec{\theta} \] | \[ \ddx{x}{\theta} = a \sec{\theta}\tan{\theta} \] | |
| Circular Functions |
\[ s = \sin{x} \] | \[ \ddx{s}{x} = \cos{x} \] |
| \[ c = \cos{x} \] | \[ \ddx{c}{x} = - \sin{x} \] | |
| \[ t = \tan{\frac{1}{2}x} \] | \[ \ddx{x}{t} = \frac{2}{1+t^2} \] | |
| Hyperbolic Functions |
\[ u = e^x \] | \[ \ddx{u}{x} = e^x \] |
| \[ s = \sinh{x} \] | \[ \ddx{s}{x} = \cosh{x} \] | |
| \[ c = \cosh{x} \] | \[ \ddx{c}{x} = \sinh{x} \] | |
| \[ t = \tanh{\frac{1}{2}x} \] | \[ \ddx{t}{x} = \frac{1}{2} \sech^2{\frac{1}{2}x} \] |
Applications of Integration
Solid of Revolution
Volume of Revolution
\[ \begin{align} V &= \pi \int_a^b{[f(x)]^2 \dx{x}}\qquad\quad = \pi \int_a^b y^2\ \dx{x} \\ V &= \pi \int_a^b{[f(x)-g(x)]^2 \dx{x}} = \pi \int_a^b (y_2-y_1)^2 \dx{x} \end{align} \]
Centroid of a Plane Area
The plane area between two continuous functions \( f(x) \lt g(x) \) on a given interval \( x \in [a,b] \) is defined as:
\[ A = \int_a^b{f(x)\dx{x}} - \int_a^b{g(x)\dx{x}} \]
The coordinates of the centroid \( (\bar{x}, \bar{y}) \) are defined by:
\[ \begin{align} \bar{x} &= \ \frac{1}{A} \int_a^b{x[f(x)-g(x)]\dx{x}} \qquad\ \ = \ \frac{1}{A} \int_a^b{x(y_2-y_1)\dx{x}} \\ \bar{y} &= \frac{1}{2A} \int_a^b{\left\{[f(x)]^2-[g(x)]^2\right\}\dx{x}} = \frac{1}{2A} \int_a^b{(y_2^2 - y_1^2)\dx{x}} \end{align} \]
Geometric Properties
Arc length of the curve:
\[ s = \int_a^b{\sqrt{1 + \left(\ddx{y}{x}\right)^2 }\dx{x}} \]
Surface area of the solid of revolution:
\[ S = 2\pi \int_a^b{y \sqrt{1 + \left(\ddx{y}{x}\right)^2 }\dx{x}} \]
Continuously Varying Quantities
Mean value
\[ \mathrm{m.v.}(f(x)) = \frac{1}{b-a} \int_a^b{f(x)\dx{x}} = \frac{1}{b-a} \int_a^b{y\ \dx{x}} \]
Root mean square value
\[ \begin{align} \left[\mathrm{r.m.s.}(f(x))\right]^2 &= \quad \frac{1}{b-a} \int_a^b{[f(x)]^2\dx{x}} = \quad\,\frac{1}{b-a} \int_a^b{y^2\dx{x}} \\ \mathrm{r.m.s.}(f(x)) &= \sqrt{\frac{1}{b-a} \int_a^b{[f(x)]^2\dx{x}}} = \sqrt{\frac{1}{b-a} \int_a^b{y^2\dx{x}}} \end{align} \]