List of Calculus Formulas Latex Code (Second Part: Integration)
rockingdingo 20221204 #Math #calculus #integrationList of Calculus Formulas Latex Code (Second Part: Integration)
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In this blog, we will summarize the latex code for basic calculus formulas, including Limits, Differentiation and Integration. For integration formulas, we will cover the topic as standard form of integration, integration of 1/x, ln(x), Exponential e^{ax}, xe^{ax}, Integration by Parts, Differentiation of an Integral, Dirac Delta Function, etc.
 1. Integration
 1.1 Standard Form
 1.2 Integration of 1/x
 1.3 Integration of ln(x)
 1.4 Integration of Exponential e^{ax}
 1.5 Integration of Exponential xe^{ax}
 1.6 Integration by Parts
 1.7 Differentiation of an Integral
 1.8 Dirac Delta Function

1.1 Standard Form
Equation
Latex Code
\int_{}^{}x^{n} \mathrm{d}x=\frac{x^{n+1}}{n+1}+c
Explanation

1.2 Integration of 1/x
Equation
Latex Code
\int_{}^{}\frac{1}{x} \mathrm{d}x=\ln(x)+c
Explanation

1.3 Integration of ln(x)
Equation
Latex Code
\int_{}^{}\ln(x) \mathrm{d}x=x(\ln(x)1) + c
Explanation

1.4 Integration of Exponential e^{ax}
Equation
Latex Code
\int_{}^{} e^{ax} \mathrm{d}x=\frac{1}{a}e^{ax} + c
Explanation

1.5 Integration of Exponential xe^{ax}
Equation
Latex Code
\int_{}^{} xe^{ax} \mathrm{d}x=e^{ax}(\frac{x}{a}\frac{1}{a^{2}}) + c
Explanation

1.6 Integration by Parts
Equation
Latex Code
\int_{a}^{b} u \mathrm{d}v = uv _{a}^{b}\int_{a}^{b} v \mathrm{d}u
Explanation
See this post for details Integration By Parts Review

1.7 Differentiation of an Integral
Equation
Latex Code
\frac{\mathrm{d}}{\mathrm{d} \alpha} \int_{a(\alpha)}^{b(\alpha)} f(x, \alpha) \mathrm{d}x=f(b,\alpha) \frac{\mathrm{d} b}{\mathrm{d} \alpha}  f(a,\alpha) \frac{\mathrm{d} a}{\mathrm{d} \alpha} + \int_{a(\alpha)}^{b(\alpha)} \frac{\partial}{\partial{\alpha}} f(x, \alpha) \mathrm{d}x
Explanation
If f(x,\alpha) is a function of x containing the parameter \alpha, and the limits of integration a(.) and b(.) are also functions of \alpha then, the above integral formula holds. See this tutorial for more detail. Integration of An Integral

1.8 Dirac Delta Function
Equation
Dirac Delta Function Definition
Latex Code
\delta(t)=0,\forall t\ne 0 \\\delta(t) \rightarrow \infty,t = 0 \\\int_{\infty}^{\infty} \delta (tt_{0}) f(t) \mathrm{d}t=f(t_{0})
Fourier Transform of Delta Function
F[\delta]=\frac{1}{2\pi}\int_{\infty}^{\infty} \exp^{i\omega t} \delta(tt_{0}) \mathrm{d}t=\frac{1}{2\pi}\exp^{i\omega t}
Inverse Transform of Delta Function
\delta (tt_{0}) = \frac{1}{2\pi}\int_{\infty}^{\infty} \exp^{i\omega (tt_{0})} \mathrm{d}\omega
Explanation
The Dirac delta function is a mathematical construct which is called a generalised function or a distribution and was originally introduced by the British theoretical physicist Paul Dirac. The function Î´(x) has the value zero everywhere except at x = 0, where its value is infinitely large and is such that its total integral is 1. This function is very useful as an approximation for a tall narrow spike function, namely an impulse. For example, to calculate the dynamics of a baseball being hit by a bat, approximating the force of the bat hitting the baseball by a delta function is a useful device. The delta function not only enables the equations to be simplified, but it also allows the motion of the baseball to be calculated by only considering the total impulse of the bat against the ball, rather than requiring the details of how the bat transferred energy to the ball. However, we note that the Dirac delta function is not strictly a function, although for many purposes it can be manipulated as such and can be formally defined as a generalised function or as a distribution that is also a measure. See this tutorial for more detail. Dirac Delta Function