Integrals

Reversing the derivative Finding the area under the function

Integral = Antiderivative

$\frac{x^3}{3}+C\to x^2$

C … constant the integration is not just a function, it is a whole class of functions. Since deriving a constant does not change the slope, there are infinitely many possibilities.

Syntax

$\int f(x)dx=F(x)+C$

$f(x)$ … integrant $F(x)$ … integration $C$ … integration constant $\int$ and $dx$ enclose the part to integrate like parenthesis

Constants

$(x+1{)}^2=x^2+2x$ $(x+1{)}^2=x^2+2x+1$ they are only different by a constant $+1$ which is still equal.

Rules

wontfix get from slides

$\int \frac{1}{x}dx=ln(|x|)+C$ … careful with the absolute bars $\int cos(x)dx=sin(x)+C$ $\int sin(x)dx=-cos(x)+C$

Example

$\int (2x^3+5)dx=\frac{x^4}{2}+5x+C$

Definite Integrals

• approximating the area under a smooth curve (not just straight lines)
• dividing function into sub-intervals
  • draw rectangle up to the function and as wide as the sub-interval
  • repeat for every rectangle
  • as the width of the sub-intervals get smaller the approximation gets more accurate, precise
• wontfix get illustration from slides
• getting smaller → infinitesimal difference $\Delta x$
• ${\lim }_{\Delta x\to \infty }$
• ${\int }_a^{b}f(x)dx$ … definite integral from $a$ to $b$ of $f(x)$
  • the area under the curve between $a$ and $b$
• ${\int }_a^{b}f(x)dx=F(b)-F(a)$
  • $F(x)$ … any of the antiderivatives of $f(x)$
  • the constant $C$ cancels out
  • $F(b)-F(a)$ is denoted as ${|}_a^{b}F(x)$
• an area below the x-axis is negative, above the x-axis is positive
• 0-definite integral → area above x-axis = area below x-axis
  • symmetric interval around 0 with odd function

Example

${\int }_0^{ln(3)}{e}^x-3dx=[e^x-3x{]}_0^{ln(3)}=-1.30$ $-1.30$ … is negative ⇒ area below the x-axis

Rules of Definite Integrals

• ${\int }_a^b=-{\int }_b^a$ … reverse limits with negative constant
• ${\int }_a^a=0$ … lower limit = upper limit ⇒ integral = 0
• ${\int }_a^c={\int }_a^b+{\int }_b^c$ … integrals can be split by limits

Naming things

• integration variable must be with the $d$
  • ${\int }_a^{b}f(\delta )d\delta$ is perfectly fine
• careful with duplicate names
  • ${\int }_x^{b}f(x)dx$ is not fine… $x$ cannot be a delimiter and the integration variable at the same time

Partial Integration

$\int f\ast gdx=F\ast g-\int F\ast g^{\prime }dx$ or $\int f^{\prime }\ast gdx=f\ast g-\int f\ast g^{\prime }dx$

Careful: partial integration “does not integrate”, it simply restructures the problem One part is integrated, another part is differentiated Can be used when one part of the function can be “differentiated away”

Example

$\int x{e}^{x}dx=x\ast e^x-\int x\ast e^x=e^x(x-1)+C$ $\int x^2{e}^{x}dx=x^2\ast e^x-2\ast \int x{e}^x=\text{see first}=e^x(x^2-2x+2)+C$

$x$ can be “differentiated away” - it does not exist inside the integral anymore and can therefore be integrated easier

more examples can be found here from my Uni Wien studies - ignore substitution exercises on this sheet - way more advanced than WU requires

Integration by Substitution

when partial integration does not work, one needs to substitute parts into new variables.

$u$ and $\frac{du}{dx}$

reverse composition

wontfix get formula from slides

Example

${\int }_0^{1/3}\frac{72x}{(9x^2+2{)}^5}dx$wontfix do example

Changing the Limits

plugging the limits into $u$wontfix to same example and changing the limits

Applications

Inverse Demand/Supply

flipping Quantity and Price → inverse functions of demand and supply

at what price is it possible to sell a particular amount of goods. ⇒ when I want so sell x quantity i have to sell at x price

Consumer Surplus

some people will be willing to buy more ⇒ Consumer Rentwontfix find reference

Area of equilibrium price $P_0$ and higher prices.

wontfix write down some formulas or excalidraw idk

wontfix check with formula from slides

Average of a Function within Interval

wontfix write down functions

Integration with $I$

recursive integration when doing partial integration and you get back the same expression in the integral one can do Integration with $I$

Example

taken from here - Aufgabe 2 - Uni Wien exercises maybe need to switch to “Preview” mode in top right $\int \sin x\cos xdx=-\cos x+\cos x+\int \cos x\ast \sin xdx$ $=-{\cos }^{2}x+\int \sin x\ast \cos xdx$ $I$ sei $\int \sin x\cos xdx$ $I=+{\cos }^{2}x-I$ $I=\frac{{\cos }^{2}x}{2}$