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Maths

Integration

Integration is a calculus operation and the inverse operation of differentiation. It finds the area under the function's line on a graph. It turns up a lot throughout physics, much to my chagrin. It may be important but I don't have to like it.

General rules

Integrating a constant

$\int a\,dx=ax+C$

Integrating x

$\int x\,dx=\frac{x^{n+1}}{n+1}+C$

Integrating constant coefficients (ax)

$\int ax\,dx=a\int x\,dx$

Adding/subtracting integrals

$\int u\pm v\,dx=\int u\,dx+\int v\,dx$

Integrating inverse functions

$\int \frac{1}{x}\,dx=\ln{x}+C$

Integrating the natural log (ln)

$\int \ln{x}\,dx= x\ln{x}-x+C$

Integrating exponentials

$\int a^x\,dx=\frac{a^x}{\ln{a}}+C$

Integrating the exponential function

$\int e^x\,dx=e^x+C$

Integrating sin(x)

$\int \sin{x}\,dx=-\cos{x}+C$

Integrating cos(x)

$\int \cos{x}\,dx=\sin{x}+C$

Integrating sec$^2$

$\int \sec^2{x}\,dx=\tan{x}+C$

Reverse Chain Rule/Integration by Substitution

Subsitution is used when functions are nested, much like chain rule. There are three main versions of this method, some of which are for specific situations:

Version 1: Find a complex part of the function of x and let it equal u. Find du/dx, and hence du, which you can substitute into the integral and solve. (Apply u to the limits when definite).

Version 2: $\int f(g(x))g'(x)\,dx=\int f(u)\,dx$. When definite: $\int_{a}^b f(g(x))g'(x)\,dx=\int_{g(a)}^{g(b)} f(u)\,dx$

Version 3: $\int{\frac{1}{ax+b}}\,dx=\frac{1}{a}\ln{|ax+b|}+c$

Integration by Parts

$\int uv'\,dx=uv-\int vu'\,dx$

or equivalently

$\int_{a}^{b}uv'\,dx=[uv]^a_b-\int_{a}^{b}vu'\,dx$

Choose the easiest to integrate part to be v’!

Symbols and Definitions

$\int dx$: Integral w.r.t. $x$.

$u$: function of x ($u(x)$), or, when subsituting, a section of a function of x.

$v$: function of x ($v(x)$)

$a$: constant

$b$: constant

$n$: constant

$C$: constant of integration

$f(x)$: function of x

$f'(x)$: derivative of a function of x ($\frac{\delta f(x)}{\delta x}$)