Here is a list of some formulas for complex integrals that may be useful:
- Integral by parts:
[
\int u \, dv = uv – \int v \, du
] - Substitution integral:
If ( x = g(t) ), then
[
\int f(x) \, dx = \int f(g(t)) g'(t) \,
dt - Integral from a fraction:
[
\int \frac{1}{x^n} \, dx = \frac{x^{-n+1}}{-n+1} + C, \quad n \neq 1
] - Integral of trigonometric functions:
[
\int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + C
]
[
\int \cos(ax) \, dx = \frac{1}{a} \sin(ax) + C
] - The exponent integral is:
[
\int e^{ax} \, dx = \frac{1}{a} e^{ax} +
C - Integral of the product of functions (substitution method):
[
\int f(g(x)) g'(x) \, dx = F(g(x)) + C
] - Integral of the root:
[
\int \sqrt{x} \, dx = \frac{2}{3} x^{3/2} + C
] - Integral of a fractional rational function (method of decomposition into simplest forms):
[
\int \frac{P(x)}{Q(x)} \, dx
]
where ( P(x) ) and ( Q(x) ) are polynomials, and decomposition into simplest forms can be used.