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# numpy.polyint

numpy.polyint(p, m=1, k=None)

  Return an antiderivative (indefinite integral) of a polynomial.

  Note

  This forms part of the old polynomial API. Since version 1.4, the new polynomial API defined in `numpy.polynomial` is preferred. A summary of the differences can be found in the transition guide.

  The returned order m antiderivative P of polynomial p satisfies \(\frac{d^m}{dx^m}P(x) = p(x)\) and is defined up to m - 1 integration constants k. The constants determine the low-order polynomial part

  \[\frac{k_{m-1}}{0!} x^0 + \ldots + \frac{k_0}{(m-1)!}x^{m-1}\]

  of P so that \(P^{(j)}(0) = k_{m-j-1}\).

  Parameters:
    p array_like or poly1d

      Polynomial to integrate. A sequence is interpreted as polynomial coefficients, see `poly1d`.

    m int, optional

      Order of the antiderivative. (Default: 1)

    k list of m scalars or scalar, optional

      Integration constants. They are given in the order of integration: those corresponding to highest-order terms come first.

      If `None` (default), all constants are assumed to be zero. If m = 1, a single scalar can be given instead of a list.

  See also

  `polyder`

    derivative of a polynomial

  `poly1d.integ`

    equivalent method

  Examples

  The defining property of the antiderivative:

      >>> import numpy as np

      >>> p = np.poly1d([1,1,1])
      >>> P = np.polyint(p)
      >>> P
       poly1d([ 0.33333333,  0.5       ,  1.        ,  0.        ]) # may vary
      >>> np.polyder(P) == p
      True

  The integration constants default to zero, but can be specified:

      >>> P = np.polyint(p, 3)
      >>> P(0)
      0.0
      >>> np.polyder(P)(0)
      0.0
      >>> np.polyder(P, 2)(0)
      0.0
      >>> P = np.polyint(p, 3, k=[6,5,3])
      >>> P
      poly1d([ 0.01666667,  0.04166667,  0.16666667,  3. ,  5. ,  3. ]) # may vary

  Note that 3 = 6 / 2!, and that the constants are given in the order of integrations. Constant of the highest-order polynomial term comes first:

      >>> np.polyder(P, 2)(0)
      6.0
      >>> np.polyder(P, 1)(0)
      5.0
      >>> P(0)
      3.0