Sinusoidal Integral

sinusoidal_integral(first_constant, second_constant, third_constant, fourth_constant, precision=4)

Generates the integral of a sinusoidal function

Parameters

• first_constant (int or float) – Vertical stretch factor of the original sine function; if zero, it will be converted to a small, non-zero decimal value (e.g., 0.0001)

• second_constant (int or float) – Horizontal stretch factor of the original sine function; if zero, it will be converted to a small, non-zero decimal value (e.g., 0.0001)

• third_constant (int or float) – Horizontal shift of the original sine function; if zero, it will be converted to a small, non-zero decimal value (e.g., 0.0001)

• fourth_constant (int or float) – Vertical shift of the original sine function; if zero, it will be converted to a small, non-zero decimal value (e.g., 0.0001)

• precision (int, default=4) – Maximum number of digits that can appear after the decimal place of the resultant roots

Raises

• TypeError – First four arguments must be integers or floats

• ValueError – Last argument must be a positive integer

Returns

• integral[‘constants’] (list of float) – Coefficients of the resultant integral

• integral[‘evaluation’] (func) – Function for evaluating the resultant integral at any float or integer argument

See also

sinusoidal_equation(), sinusoidal_derivatives(), sinusoidal_roots(), sinusoidal_model()

Notes

• Standard form of a sinusoidal function: \(f(x) = a\cdot{\sin(b\cdot(x - c))} + d\)

• Integral of a sinusoidal function: \(F(x) = -\frac{a}{b}\cdot{\cos(b\cdot(x - c))} + d\cdot{x}\)

• Basic Properties of Indefinite Integrals

• Basic Integration Forumulas

• Substitution Rule

Examples

Import sinusoidal_integral function from regressions library

>>> from regressions.analyses.integrals.sinusoidal import sinusoidal_integral

Generate the integral of a sinusoidal function with coefficients 2, 3, 5, and 7, then display its coefficients

>>> integral_constants = sinusoidal_integral(2, 3, 5, 7)
>>> print(integral_constants['constants'])
[-0.6667, 3.0, 5.0, 7.0]

Generate the integral of a sinusoidal function with coefficients 7, -5, -3, and 2, then evaluate its integral at 10

>>> integral_evaluation = sinusoidal_integral(7, -5, -3, 2)
>>> print(integral_evaluation['evaluation'](10))
19.2126

Generate the integral of a sinusoidal function with all inputs set to 0, then display its coefficients

>>> integral_zeroes = sinusoidal_integral(0, 0, 0, 0)
>>> print(integral_zeroes['constants'])
[-1.0, 0.0001, 0.0001, 0.0001]