class_docstr: Univariate spline in the B-spline basis.
  
  .. math::
      S(x) = \sum_{j=0}^{n-1} c_j  B_{j, k; t}(x)
  
  where :math:`B_{j, k; t}` are B-spline basis functions of degree `k`
  and knots `t`.
  
  Parameters
  ----------
  t : ndarray, shape (n+k+1,)
      knots
  c : ndarray, shape (>=n, ...)
      spline coefficients
  k : int
      B-spline degree
  extrapolate : bool or 'periodic', optional
      whether to extrapolate beyond the base interval, ``t[k] .. t[n]``,
      or to return nans.
      If True, extrapolates the first and last polynomial pieces of b-spline
      functions active on the base interval.
      If 'periodic', periodic extrapolation is used.
      Default is True.
  axis : int, optional
      Interpolation axis. Default is zero.
  
  Attributes
  ----------
  t : ndarray
      knot vector
  c : ndarray
      spline coefficients
  k : int
      spline degree
  extrapolate : bool
      If True, extrapolates the first and last polynomial pieces of b-spline
      functions active on the base interval.
  axis : int
      Interpolation axis.
  tck : tuple
      A read-only equivalent of ``(self.t, self.c, self.k)``
  
  Notes
  -----
  B-spline basis elements are defined via
  
  .. math::
      B_{i, 0}(x) = 1, \textrm{if $t_i \le x < t_{i+1}$, otherwise $0$,}
  
      B_{i, k}(x) = \frac{x - t_i}{t_{i+k} - t_i} B_{i, k-1}(x)
               + \frac{t_{i+k+1} - x}{t_{i+k+1} - t_{i+1}} B_{i+1, k-1}(x)
  
  **Implementation details**
  
  - At least ``k+1`` coefficients are required for a spline of degree `k`,
    so that ``n >= k+1``. Additional coefficients, ``c[j]`` with
    ``j > n``, are ignored.
  
  - B-spline basis elements of degree `k` form a partition of unity on the
    *base interval*, ``t[k] <= x <= t[n]``.
  
  - Based on [1]_ and [2]_
  
  .. seealso:: :class:`scipy.interpolate.BSpline`
  
  References
  ----------
  .. [1] Tom Lyche and Knut Morken, Spline methods,
      http://www.uio.no/studier/emner/matnat/ifi/INF-MAT5340/v05/undervisningsmateriale/
  .. [2] Carl de Boor, A practical guide to splines, Springer, 2001.
docstr: Compute a definite integral of the spline.
  
  Parameters
  ----------
  a : float
      Lower limit of integration.
  b : float
      Upper limit of integration.
  extrapolate : bool or 'periodic', optional
      whether to extrapolate beyond the base interval,
      ``t[k] .. t[-k-1]``, or take the spline to be zero outside of the
      base interval. If 'periodic', periodic extrapolation is used.
      If None (default), use `self.extrapolate`.
  
  Returns
  -------
  I : array_like
      Definite integral of the spline over the interval ``[a, b]``.
docstr_tok: ["Compute","a","definite","integral","of","the","spline",".","Parameters","--","--","--","--","--","a",":","float","Lower","limit","of","integration",".","b",":","float","Upper","limit","of","integration",".","extrapolate",":","bool","or","'periodic","'",",","optional","whether","to","extrapolate","beyond","the","base","interval",",","``","t","[","k","]","..","t","[","-k-1","]","``",",","or","take","the","spline","to","be","zero","outside","of","the","base","interval",".","If","'periodic","'",",","periodic","extrapolation","is","used",".","If","None","(","default",")",",","use","`","self.extrapolate","`",".","Returns","--","--","--","-","I",":","array_like","Definite","integral","of","the","spline","over","the","interval","``","[","a",",","b","]","``","."]
code: def integrate(self, a, b, extrapolate=None):
      """
      Compute a definite integral of the spline.
  
      Parameters
      ----------
      a : float
          Lower limit of integration.
      b : float
          Upper limit of integration.
      extrapolate : bool or 'periodic', optional
          whether to extrapolate beyond the base interval,
          ``t[k] .. t[-k-1]``, or take the spline to be zero outside of the
          base interval. If 'periodic', periodic extrapolation is used.
          If None (default), use `self.extrapolate`.
  
      Returns
      -------
      I : array_like
          Definite integral of the spline over the interval ``[a, b]``.
      """
      if extrapolate is None:
          extrapolate = self.extrapolate
  
      # Prepare self.t and self.c.
      self._ensure_c_contiguous()
  
      # Swap integration bounds if needed.
      sign = 1
      if b < a:
          a, b = b, a
          sign = -1
      n = self.t.size - self.k - 1
  
      if extrapolate != "periodic" and not extrapolate:
          # Shrink the integration interval, if needed.
          a = max(a, self.t[self.k].item())
          b = min(b, self.t[n].item())
  
          # if self.c.ndim == 1:
          #     # Fast path: use FITPACK's routine
          #     # (cf _fitpack_impl.splint).
          #     integral = splint(a, b, self.tck)
          #     return integral * sign
  
      out = cupy.empty(
          (2, int(np.prod(self.c.shape[1:]))), dtype=self.c.dtype
      )
  
      # Compute the antiderivative.
      c = self.c
      ct = len(self.t) - len(c)
      if ct > 0:
          c = cupy.r_[c, cupy.zeros((ct,) + c.shape[1:])]
      ta, ca, ka = splantider((self.t, c, self.k), 1)
  
      if extrapolate == "periodic":
          # Split the integral into the part over period (can be several
          # of them) and the remaining part.
  
          ts, te = self.t[self.k], self.t[n]
          period = te - ts
          interval = b - a
          n_periods, left = divmod(interval, period)
  
          if n_periods > 0:
              # Evaluate the difference of antiderivatives.
              x = cupy.asarray([ts, te], dtype=cupy.float_)
              _evaluate_spline(
                  ta, ca.reshape(ca.shape[0], -1), ka, x, 0, False, out
              )
              integral = out[1] - out[0]
              integral *= n_periods
          else:
              integral = cupy.zeros(
                  (1, int(np.prod(self.c.shape[1:]))),
                  dtype=self.c.dtype,
              )
  
          # Map a to [ts, te], b is always a + left.
          a = ts + (a - ts) % period
          b = a + left
  
          # If b <= te then we need to integrate over [a, b], otherwise
          # over [a, te] and from xs to what is remained.
          if b <= te:
              x = cupy.asarray([a, b], dtype=cupy.float_)
              _evaluate_spline(
                  ta, ca.reshape(ca.shape[0], -1), ka, x, 0, False, out
              )
              integral += out[1] - out[0]
          else:
              x = cupy.asarray([a, te], dtype=cupy.float_)
              _evaluate_spline(
                  ta, ca.reshape(ca.shape[0], -1), ka, x, 0, False, out
              )
              integral += out[1] - out[0]
  
              x = cupy.asarray([ts, ts + b - te], dtype=cupy.float_)
              _evaluate_spline(
                  ta, ca.reshape(ca.shape[0], -1), ka, x, 0, False, out
              )
              integral += out[1] - out[0]
      else:
          # Evaluate the difference of antiderivatives.
          x = cupy.asarray([a, b], dtype=cupy.float_)
          _evaluate_spline(
              ta,
              ca.reshape(ca.shape[0], -1),
              ka,
              x,
              0,
              extrapolate,
              out,
          )
          integral = out[1] - out[0]
  
      integral *= sign
      return integral.reshape(ca.shape[1:])
  
imports: import operator
  import cupy
  from cupy._core import internal
  from cupy._core._scalar import get_typename
  from cupyx.scipy.sparse import csr_matrix
  import numpy
label_desc: Use image node_id 11 for calling the BSpline obj's underlying member method code with example usage: obj.integrate(a, b, extrapolate) and returns: integral