Sine_Power_Integral — Special

Evaluates the sine power integral.

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View versions (2)

Interface

#include <codecogs/maths/special/sine_power_integral.h>

using namespace Maths::Special;

double sine_power_integral(double a, double b, int n)

This function evaluates the sine power integral with given parameters, defined by

$$\mathrm{sinepower}(a, b, n) = \int_a^b \sin(t)^n \mathrm{d}t$$

(1)

The algorithm uses the fact that

$$\int \sin^n(t) \mathrm{d}t = \frac{1}{n} \left( \sin^{n-1}(t) \cos(t) + (n-1) \int \sin^{n-2}(t) \mathrm{d}t \right)$$

(2)

Example 1

#include <codecogs/maths/special/sine_power_integral.h>
#include <iostream>
#include <iomanip>

int main()
{
  std::cout << std::setprecision(10);
  std::cout << "For lower and upper bounds a = 0 and b = 10, " << std::endl;
  std::cout << "and 0 <= n < 10, the value of the integral is" << std::endl;
  for (int n = 0; n < 10; n++)
  {
    std::cout << "n = " << n << " : ";
    std::cout << Maths::Special::sine_power_integral(0, 10, n) << std::endl;
  }
  return 0;
}

Output

For lower and upper bounds a = 0 and b = 10,
and 0 <= n < 10, the value of the integral is
n = 0 : 10
n = 1 : 1.839071529
n = 2 : 4.771763687
n = 3 : 1.308824601
n = 4 : 3.545048474
n = 5 : 1.061758823
n = 6 : 2.947543192
n = 7 : 0.913186379
n = 8 : 2.577621119
n = 9 : 0.8124365163

Parameters

the lower limit of integration

the upper limit of integration

the power of the sine function

Returns

An approximation of the sine power integral.

References

John Burkardt's library of statistical C++ routines, http://www.csit.fsu.edu/~burkardt/cpp_src/prob/prob.html

Interactive Calculator

Result –