Derivative Of Integrals

Take note of the limits here. The upper limit must be the variable you are differentiating by, and the function in the integral must not. \[\frac{d}{dx}\int^{x}_{a}f(t)dt=f(x)\] Proof \[ \begin{align} &\frac{d}{dx}\int^x_af(t)dt=\frac{d}{dt}\left [F(t)\right ]^x_a\\\\ &=\frac{d}{dx}F(x)-\frac{d}{dx}F(a)\\\\ &=f(x)-0\\\\ &=f(x) \end{align} \]

Trig-Substitution

Three forms and their respective substitutions:

Form	Substitution
\(\int\sqrt{a^2-x^2}dx\)	\(x=a\sin\theta\)
\(\int\sqrt{a^2+x^2}dx\)	\(x=a\tan\theta\)
\(\int\sqrt{x^2-a^2}dx\)	\(x=a\sec\theta\)

Partial Fractions

1. Unrepeated linear factor \[\frac{A}{ax+b}\]
2. Repeated linear factor \[\frac{B}{(ax+b)^m},\;\;m\geq2\]
3. Unrepeated quadratic factor \[\frac{Cx+D}{ax^2+bx+c}\]
4. Repeated quadratic factor \[\frac{Ex+F}{(ax^2+bx+c)^n},\;\;n\geq2\]

Improper Integrals

Type I - Infinite range of integration

We can treat type I improper integrals as limits: \[\int^\infty_af(x)dx=\lim_{b\to\infty}\int_a^bf(x)dx=l\] If the limit \(l\) is finite, the integral is said to converge and be equal to \(l\). If \(l\to\pm\infty\) (or does not exist) as \(b\to\infty\), then the integral is said to diverge. Note: the following is also possible \[\int^\infty_{-\infty}f(x)dx=\lim_{a\to-\infty}\int_a^0f(x)dx+\lim_{b\to\infty}\int_0^bf(x)dx\] …Which, if either or both of these limits diverge, then the original expression diverges also.

Simple convergence test

For a integral \(\int^\infty_af(x)dx\): If for a large \(x\), \(f(x)=\frac{g(x)}{x^n}\) is such that \(g(x)\) is non zero:

1. \(|g(x)| < C\) (arbitrary constant)
2. n>1

Then the integral is convergent.

Type II - Discontinuous Integrand

If for integral: \(\int^b_af(x)dx\), the range of \(a\;\&\;b\) are finite, but \(f(x)\to\pm\infty\) as \(x\to a\), \(x=a\) is known as a singularity point. Such we can define the improper integral: \[\int^b_af(x)dx=\lim_{\epsilon\to0}\int^b_{a+\epsilon}f(x)dx=l\] If \(l\) is finite, the integral will converge and be equal to \(l\), else it will either be infinite or not exist and in which case will be said to diverge.

Multiple Integrals

Transforming Coordinate Systems

When translating from cartesian coordinates to polar, cylindrical or spherical coordinates, we must multiply the function by the Jacobian. \[\begin{array}{l} \textrm{polar:}\iint dxdy\implies\iint r\cdot drd\theta\\ \textrm{cylindrical:}\iiint dxdydz\implies\iiint r\cdot dzdrd\theta\\ \textrm{Spherical:}\iiint dxdydz\implies \iiint\rho\sin^2(\phi)\cdot d\rho d\theta d\phi \end{array}\]

Special Functions

Heaviside Function

(Step function) \[H(x)\equiv \Theta(x)\equiv v(x)\] It is related to the modulus function... \[|x|=x\cdot(H(x)-H(-x))\]



This is useful in many applications of physics, such as momentum changes when a ball hits a wall.

Delta Function

This one is harder to understand... Definition: \[\delta(t)=\frac{dH(t)}{dt}\qquad:\qquad\int\delta(t)dt=1\]

Line Integrals

Definition: Let \(f(x,y)\;\;\&\;\;g(x,y)\) be single values functions. \[I_{line}=\int_{(AB)}(f(x,y)dx\;+\;g(x,y)dy)\] This way is a bit difficult to apply, so it is generally better to use parameterisation. Length of a curve: \[L=\int_Cf(x,y)ds\qquad\qquad ds=\sqrt{\left (\frac{dx}{dt} \right)^2 + \left (\frac{dy}{dt} \right)^2}dt\] Where the limits are the bounds of \(t\).