### 19.2 Functions and Variables for Integration

Function: changevar (expr, f(x,y), y, x)

  Makes the change of variable given by `f(x,y) = 0` in all integrals occurring in expr with integration with respect to x. The new variable is y.

  The change of variable can also be written `f(x) = g(y)`.

      (%i1) assume(a > 0)$

      (%i2) 'integrate (%e**sqrt(a*y), y, 0, 4);
                            4
                           /
                           [    sqrt(a) sqrt(y)
      (%o2)                I  %e                dy
                           ]
                           /
                            0

      (%i3) changevar (%, y-z^2/a, z, y);
                            0
                           /
                           [                abs(z)
                         2 I            z %e       dz
                           ]
                           /
                            - 2 sqrt(a)
      (%o3)            - ----------------------------
                                      a

  An expression containing a noun form, such as the instances of `'integrate` above, may be evaluated by `ev` with the `nouns` flag. For example, the expression returned by `changevar` above may be evaluated by `ev (%o3, nouns)`.

  `changevar` may also be used to changes in the indices of a sum or product. However, it must be realized that when a change is made in a sum or product, this change must be a shift, i.e., `i = j+ ...`, not a higher degree function. E.g.,

      (%i4) sum (a[i]*x^(i-2), i, 0, inf);
                               inf
                               ====
                               \         i - 2
      (%o4)                     >    a  x
                               /      i
                               ====
                               i = 0

      (%i5) changevar (%, i-2-n, n, i);
                              inf
                              ====
                              \               n
      (%o5)                    >      a      x
                              /        n + 2
                              ====
                              n = - 2

  ·

  Categories: Integral calculus

Function: dblint (f, r, s, a, b)

  A double-integral routine which was written in top-level Maxima and then translated and compiled to machine code. Use `load ("dblint")` to access this package. It uses the Simpson’s rule method in both the x and y directions to calculate

      /b /s(x)
      |  |
      |  |    f(x,y) dy dx
      |  |
      /a /r(x)

  The function f must be a translated or compiled function of two variables, and r and s must each be a translated or compiled function of one variable, while a and b must be floating point numbers. The routine has two global variables which determine the number of divisions of the x and y intervals: `dblint_x` and `dblint_y`, both of which are initially 10, and can be changed independently to other integer values (there are `2*dblint_x+1` points computed in the x direction, and `2*dblint_y+1` in the y direction). The routine subdivides the X axis and then for each value of X it first computes `r(x)` and `s(x)`; then the Y axis between `r(x)` and `s(x)` is subdivided and the integral along the Y axis is performed using Simpson’s rule; then the integral along the X axis is done using Simpson’s rule with the function values being the Y-integrals. This procedure may be numerically unstable for a great variety of reasons, but is reasonably fast: avoid using it on highly oscillatory functions and functions with singularities (poles or branch points in the region). The Y integrals depend on how far apart `r(x)` and `s(x)` are, so if the distance `s(x) - r(x)` varies rapidly with X, there may be substantial errors arising from truncation with different step-sizes in the various Y integrals. One can increase `dblint_x` and `dblint_y` in an effort to improve the coverage of the region, at the expense of computation time. The function values are not saved, so if the function is very time-consuming, you will have to wait for re-computation if you change anything (sorry). It is required that the functions f, r, and s be either translated or compiled prior to calling `dblint`. This will result in orders of magnitude speed improvement over interpreted code in many cases!

  `demo ("dblint")` executes a demonstration of `dblint` applied to an example problem.

  ·

  Categories: Integral calculus

Function: defint (expr, x, a, b)

  Attempts to compute a definite integral. `defint` is called by `integrate` when limits of integration are specified, i.e., when `integrate` is called as `integrate (expr, x, a, b)`. Thus from the user’s point of view, it is sufficient to call `integrate`.

  `defint` returns a symbolic expression, either the computed integral or the noun form of the integral. See `quad_qag` and related functions for numerical approximation of definite integrals.

  ·

  Categories: Integral calculus

Option variable: erfflag

  Default value: `true`

  When `erfflag` is `false`, prevents `risch` from introducing the `erf` function in the answer if there were none in the integrand to begin with.

  ·

  Categories: Integral calculus

Function: ilt (expr, s, t)

  Computes the inverse Laplace transform of expr with respect to s and parameter t. expr must be a ratio of polynomials whose denominator has only linear and quadratic factors; there is an extension of `ilt`, called `pwilt` (Piece-Wise Inverse Laplace Transform) that handles several other cases where `ilt` fails.

  By using the functions `laplace` and `ilt` together with the `solve` or `linsolve` functions the user can solve a single differential or convolution integral equation or a set of them.

      (%i1) 'integrate (sinh(a*x)*f(t-x), x, 0, t) + b*f(t) = t**2;
                    t
                   /
                   [                                    2
      (%o1)        I  f(t - x) sinh(a x) dx + b f(t) = t
                   ]
                   /
                    0

      (%i2) laplace (%, t, s);
                                     a laplace(f(t), t, s)   2
      (%o2)  b laplace(f(t), t, s) + --------------------- = --
                                             2    2           3
                                            s  - a           s

      (%i3) linsolve ([%], ['laplace(f(t), t, s)]);
                                              2      2
                                           2 s  - 2 a
      (%o3)     [laplace(f(t), t, s) = --------------------]
                                          5         2     3
                                       b s  + (a - a  b) s

      (%i4) ilt (rhs (first (%)), s, t);
      Is  a b (a b - 1)  positive, negative, or zero?

      pos;
                     sqrt(a b (a b - 1)) t
              2 cosh(---------------------)       2
                               b               a t
      (%o4) - ----------------------------- + -------
                    3  2      2               a b - 1
                   a  b  - 2 a  b + a

                                                             2
                                                   + ------------------
                                                      3  2      2
                                                     a  b  - 2 a  b + a

  ·

  Categories: Laplace transform

Option variable: intanalysis

  Default value: `true`

  When `true`, definite integration tries to find poles in the integrand in the interval of integration. If there are, then the integral is evaluated appropriately as a principal value integral. If intanalysis is `false`, this check is not performed and integration is done assuming there are no poles.

  See also `ldefint`.

  Examples:

  Maxima can solve the following integrals, when `intanalysis` is set to `false`:

      (%i1) integrate(1/(sqrt(x)+1),x,0,1);
                                      1
                                     /
                                     [       1
      (%o1)                          I  ----------- dx
                                     ]  sqrt(x) + 1
                                     /
                                      0

      (%i2) integrate(1/(sqrt(x)+1),x,0,1),intanalysis:false;
      (%o2)                            2 - 2 log(2)

      (%i3) integrate(cos(a)/sqrt((tan(a))^2 +1),a,-%pi/2,%pi/2);
      The number 1 isn't in the domain of atanh
       -- an error. To debug this try: debugmode(true);

      (%i4) intanalysis:false$
      (%i5) integrate(cos(a)/sqrt((tan(a))^2+1),a,-%pi/2,%pi/2);
                                            %pi
      (%o5)                                 ---
                                             2

  ·

  Categories: Integral calculus

Function: integrate
`integrate` (expr, x)
`integrate` (expr, x, a, b)

  Attempts to symbolically compute the integral of expr with respect to x. `integrate (expr, x)` is an indefinite integral, while `integrate (expr, x, a, b)` is a definite integral, with limits of integration a and b. The limits should not contain x, although `integrate` does not enforce this restriction. a need not be less than b. If b is equal to a, `integrate` returns zero.

  See `quad_qag` and related functions for numerical approximation of definite integrals. See `residue` for computation of residues (complex integration). See `antid` for an alternative means of computing indefinite integrals.

  The integral (an expression free of `integrate`) is returned if `integrate` succeeds. Otherwise the return value is the noun form of the integral (the quoted operator `'integrate`) or an expression containing one or more noun forms. The noun form of `integrate` is displayed with an integral sign.

  In some circumstances it is useful to construct a noun form by hand, by quoting `integrate` with a single quote, e.g., `'integrate (expr, x)`. For example, the integral may depend on some parameters which are not yet computed. The noun may be applied to its arguments by `ev (i, nouns)` where i is the noun form of interest.

  `integrate` handles definite integrals separately from indefinite, and employs a range of heuristics to handle each case. Special cases of definite integrals include limits of integration equal to zero or infinity (`inf` or `minf`), trigonometric functions with limits of integration equal to zero and `%pi` or `2 %pi`, rational functions, integrals related to the definitions of the `beta` and `psi` functions, and some logarithmic and trigonometric integrals. Processing rational functions may include computation of residues. If an applicable special case is not found, an attempt will be made to compute the indefinite integral and evaluate it at the limits of integration. This may include taking a limit as a limit of integration goes to infinity or negative infinity; see also `ldefint`.

  Special cases of indefinite integrals include trigonometric functions, exponential and logarithmic functions, and rational functions. `integrate` may also make use of a short table of elementary integrals.

  `integrate` may carry out a change of variable if the integrand has the form `f(g(x)) * diff(g(x), x)`. `integrate` attempts to find a subexpression `g(x)` such that the derivative of `g(x)` divides the integrand. This search may make use of derivatives defined by the `gradef` function. See also `changevar` and `antid`.

  If none of the preceding heuristics find the indefinite integral, the Risch algorithm is executed. The flag `risch` may be set as an `evflag`, in a call to `ev` or on the command line, e.g., `ev (integrate (expr, x), risch)` or `integrate (expr, x), risch`. If `risch` is present, `integrate` calls the `risch` function without attempting heuristics first. See also `risch`.

  `integrate` works only with functional relations represented explicitly with the `f(x)` notation. `integrate` does not respect implicit dependencies established by the `depends` function.

  `integrate` may need to know some property of a parameter in the integrand. `integrate` will first consult the `assume` database, and, if the variable of interest is not there, `integrate` will ask the user. Depending on the question, suitable responses are `yes;` or `no;`, or `pos;`, `zero;`, or `neg;`.

  `integrate` is not, by default, declared to be linear. See `declare` and `linear`.

  `integrate` attempts integration by parts only in a few special cases.

  Examples:

  - Elementary indefinite and definite integrals.

        (%i1) integrate (sin(x)^3, x);
                                   3
                                cos (x)
        (%o1)                   ------- - cos(x)
                                   3

        (%i2) integrate (x/ sqrt (b^2 - x^2), x);
                                         2    2
        (%o2)                    - sqrt(b  - x )

        (%i3) integrate (cos(x)^2 * exp(x), x, 0, %pi);
                                       %pi
                                   3 %e      3
        (%o3)                      ------- - -
                                      5      5

        (%i4) integrate (x^2 * exp(-x^2), x, minf, inf);
                                    sqrt(%pi)
        (%o4)                       ---------
                                        2

  - Use of `assume` and interactive query.

        (%i1) assume (a > 1)$

        (%i2) integrate (x**a/(x+1)**(5/2), x, 0, inf);
            2 a + 2
        Is  -------  an integer?
               5

        no;
        Is  2 a - 3  positive, negative, or zero?

        neg;
                                           3
        (%o2)                  beta(a + 1, - - a)
                                           2

  - Change of variable. There are two changes of variable in this example: one using a derivative established by `gradef`, and one using the derivation `diff(r(x))` of an unspecified function `r(x)`.

        (%i3) gradef (q(x), sin(x**2));
        (%o3)                         q(x)

        (%i4) diff (log (q (r (x))), x);
                              d               2
                             (-- (r(x))) sin(r (x))
                              dx
        (%o4)                ----------------------
                                    q(r(x))

        (%i5) integrate (%, x);
        (%o5)                     log(q(r(x)))

  - Return value contains the `'integrate` noun form. In this example, Maxima can extract one factor of the denominator of a rational function, but cannot factor the remainder or otherwise find its integral. `grind` shows the noun form `'integrate` in the result. See also `integrate_use_rootsof` for more on integrals of rational functions.

        (%i1) expand ((x-4) * (x^3+2*x+1));
                            4      3      2
        (%o1)              x  - 4 x  + 2 x  - 7 x - 4

        (%i2) integrate (1/%, x);
                                      /  2
                                      [ x  + 4 x + 18
                                      I ------------- dx
                                      ]  3
                         log(x - 4)   / x  + 2 x + 1
        (%o2)            ---------- - ------------------
                             73               73

        (%i3) grind (%);
        log(x-4)/73-('integrate((x^2+4*x+18)/(x^3+2*x+1),x))/73$

  - Defining a function in terms of an integral. The body of a function is not evaluated when the function is defined. Thus the body of `f_1` in this example contains the noun form of `integrate`. The quote-quote operator `''` causes the integral to be evaluated, and the result becomes the body of `f_2`.

        (%i1) f_1 (a) := integrate (x^3, x, 1, a);
                                             3
        (%o1)           f_1(a) := integrate(x , x, 1, a)

        (%i2) ev (f_1 (7), nouns);
        (%o2)                          600

        (%i3) /* Note parentheses around integrate(...) here */
              f_2 (a) := ''(integrate (x^3, x, 1, a));
                                           4
                                          a    1
        (%o3)                   f_2(a) := -- - -
                                          4    4

        (%i4) f_2 (7);
        (%o4)                          600

  ·

  Categories: Integral calculus

System variable: integration_constant

  Default value: `%c`

  When a constant of integration is introduced by indefinite integration of an equation, the name of the constant is constructed by concatenating `integration_constant` and `integration_constant_counter`.

  `integration_constant` may be assigned any symbol.

  Examples:

      (%i1) integrate (x^2 = 1, x);
                                 3
                                x
      (%o1)                     -- = x + %c1
                                3

      (%i2) integration_constant : 'k;
      (%o2)                           k

      (%i3) integrate (x^2 = 1, x);
                                  3
                                 x
      (%o3)                      -- = x + k2
                                 3

  ·

  Categories: Integral calculus

System variable: integration_constant_counter

  Default value: 0

  When a constant of integration is introduced by indefinite integration of an equation, the name of the constant is constructed by concatenating `integration_constant` and `integration_constant_counter`.

  `integration_constant_counter` is incremented before constructing the next integration constant.

  Examples:

      (%i1) integrate (x^2 = 1, x);
                                 3
                                x
      (%o1)                     -- = x + %c1
                                3

      (%i2) integrate (x^2 = 1, x);
                                 3
                                x
      (%o2)                     -- = x + %c2
                                3

      (%i3) integrate (x^2 = 1, x);
                                 3
                                x
      (%o3)                     -- = x + %c3
                                3

      (%i4) reset (integration_constant_counter);
      (%o4)            [integration_constant_counter]

      (%i5) integrate (x^2 = 1, x);
                                 3
                                x
      (%o5)                     -- = x + %c1
                                3

  ·

  Categories: Integral calculus

Option variable: integrate_use_rootsof

  Default value: `false`

  When `integrate_use_rootsof` is `true` and the denominator of a rational function cannot be factored, `integrate` returns the integral in a form which is a sum over the roots (not yet known) of the denominator.

  For example, with `integrate_use_rootsof` set to `false`, `integrate` returns an unsolved integral of a rational function in noun form:

      (%i1) integrate_use_rootsof: false$

      (%i2) integrate (1/(1+x+x^5), x);
              /  2
              [ x  - 4 x + 5
              I ------------ dx                            2 x + 1
              ]  3    2                2            5 atan(-------)
              / x  - x  + 1       log(x  + x + 1)          sqrt(3)
      (%o2)   ----------------- - --------------- + ---------------
                      7                 14             7 sqrt(3)

  Now we set the flag to be true and the unsolved part of the integral will be expressed as a summation over the roots of the denominator of the rational function:

      (%i3) integrate_use_rootsof: true$

      (%i4) integrate (1/(1+x+x^5), x);
            ====        2
            \       (%r4  - 4 %r4 + 5) log(x - %r4)
             >      -------------------------------
            /                    2
            ====            3 %r4  - 2 %r4
                              3      2
            %r4 in rootsof(%r4  - %r4  + 1, %r4)
      (%o4) ----------------------------------------------------------
                     7

                                                            2 x + 1
                                        2            5 atan(-------)
                                   log(x  + x + 1)          sqrt(3)
                                 - --------------- + ---------------
                                         14             7 sqrt(3)

  Alternatively the user may compute the roots of the denominator separately, and then express the integrand in terms of these roots, e.g., `1/((x - a)*(x - b)*(x - c))` or `1/((x^2 - (a+b)*x + a*b)*(x - c))` if the denominator is a cubic polynomial. Sometimes this will help Maxima obtain a more useful result.

  ·

  Categories: Integral calculus

Function: ldefint (expr, x, a, b)

  Attempts to compute the definite integral of expr by using `limit` to evaluate the indefinite integral of expr with respect to x at the upper limit b and at the lower limit a. If it fails to compute the definite integral, `ldefint` returns an expression containing limits as noun forms.

  `ldefint` is not called from `integrate`, so executing `ldefint (expr, x, a, b)` may yield a different result than `integrate (expr, x, a, b)`. `ldefint` always uses the same method to evaluate the definite integral, while `integrate` may employ various heuristics and may recognize some special cases.

  ·

  Categories: Integral calculus

Function: pwilt (expr, s, t)

  Computes the inverse Laplace transform of expr with respect to s and parameter t. Unlike `ilt`, `pwilt` is able to return piece-wise and periodic functions and can also handle some cases with polynomials of degree greater than 3 in the denominator.

  Two examples where `ilt` fails:

      (%i1) pwilt (exp(-s)*s/(s^3-2*s-s+2), s, t);
                                             t - 1       - 2 (t - 1)
                                   (t - 1) %e        2 %e
      (%o1)         hstep(t - 1) (--------------- - ---------------)
                                          3                 9

      (%i2) pwilt ((s^2+2)/(s^2-1), s, t);
                                               t       - t
                                           3 %e    3 %e
      (%o2)                    delta(t) + ----- - -------
                                             2        2

  ·

  Categories: Laplace transform

Function: potential (givengradient)

  The calculation makes use of the global variable `potentialzeroloc[0]` which must be `nonlist` or of the form

      [indeterminatej=expressionj, indeterminatek=expressionk, ...]

  the former being equivalent to the nonlist expression for all right-hand sides in the latter. The indicated right-hand sides are used as the lower limit of integration. The success of the integrations may depend upon their values and order. `potentialzeroloc` is initially set to 0.

Function: residue (expr, z, z_0)

  Computes the residue in the complex plane of the expression expr when the variable z assumes the value z_0. The residue is the coefficient of `(z - z_0)^(-1)` in the Laurent series for expr.

      (%i1) residue (s/(s**2+a**2), s, a*%i);
                                      1
      (%o1)                           -
                                      2

      (%i2) residue (sin(a*x)/x**4, x, 0);
                                       3
                                      a
      (%o2)                         - --
                                      6

  ·

  Categories: Integral calculus ·Complex variables

Function: risch (expr, x)

  Integrates expr with respect to x using the transcendental case of the Risch algorithm. (The algebraic case of the Risch algorithm has not been implemented.) This currently handles the cases of nested exponentials and logarithms which the main part of `integrate` can’t do. `integrate` will automatically apply `risch` if given these cases.

  `erfflag`, if `false`, prevents `risch` from introducing the `erf` function in the answer if there were none in the integrand to begin with.

      (%i1) risch (x^2*erf(x), x);
                                                              2
                   3                      2                - x
              %pi x  erf(x) + (sqrt(%pi) x  + sqrt(%pi)) %e
      (%o1)   -------------------------------------------------
                                    3 %pi

      (%i2) diff(%, x), ratsimp;
                                   2
      (%o2)                       x  erf(x)

  ·

  Categories: Integral calculus

Function: tldefint (expr, x, a, b)

  Equivalent to `ldefint` with `tlimswitch` set to `true`.

  ·

  Categories: Integral calculus