Symbolic Integration

class sage.symbolic.integration.integral.DefiniteIntegral

Bases: sage.symbolic.function.BuiltinFunction

The symbolic function representing a definite integral.

EXAMPLES:

sage: from sage.symbolic.integration.integral import definite_integral
sage: definite_integral(sin(x),x,0,pi)
2
class sage.symbolic.integration.integral.IndefiniteIntegral

Bases: sage.symbolic.function.BuiltinFunction

Class to represent an indefinite integral.

EXAMPLES:

sage: from sage.symbolic.integration.integral import indefinite_integral
sage: indefinite_integral(log(x), x) #indirect doctest
x*log(x) - x
sage: indefinite_integral(x^2, x)
1/3*x^3
sage: indefinite_integral(4*x*log(x), x)
2*x^2*log(x) - x^2
sage: indefinite_integral(exp(x), 2*x)
2*e^x
sage.symbolic.integration.integral.integral(expression, v=None, a=None, b=None, algorithm=None, hold=False)

Return the indefinite integral with respect to the variable \(v\), ignoring the constant of integration. Or, if endpoints \(a\) and \(b\) are specified, returns the definite integral over the interval \([a, b]\).

If self has only one variable, then it returns the integral with respect to that variable.

If definite integration fails, it could be still possible to evaluate the definite integral using indefinite integration with the Newton - Leibniz theorem (however, the user has to ensure that the indefinite integral is continuous on the compact interval \([a,b]\) and this theorem can be applied).

INPUT:

  • v - a variable or variable name. This can also be a tuple of the variable (optional) and endpoints (i.e., (x,0,1) or (0,1)).

  • a - (optional) lower endpoint of definite integral

  • b - (optional) upper endpoint of definite integral

  • algorithm - (default: ‘maxima’) one of

    • ‘maxima’ - use maxima (the default)

    • ‘sympy’ - use sympy (also in Sage)

    • ‘mathematica_free’ - use http://integrals.wolfram.com/

    • ‘fricas’ - use FriCAS (the optional fricas spkg has to be installed)

    • ‘giac’ - use Giac

To prevent automatic evaluation use the hold argument.

See also

To integrate a polynomial over a polytope, use the optional latte_int package sage.geometry.polyhedron.base.Polyhedron_base.integrate().

EXAMPLES:

sage: x = var('x')
sage: h = sin(x)/(cos(x))^2
sage: h.integral(x)
1/cos(x)

sage: f = x^2/(x+1)^3
sage: f.integral(x)
1/2*(4*x + 3)/(x^2 + 2*x + 1) + log(x + 1)

sage: f = x*cos(x^2)
sage: f.integral(x, 0, sqrt(pi))
0
sage: f.integral(x, a=-pi, b=pi)
0

sage: f(x) = sin(x)
sage: f.integral(x, 0, pi/2)
1

The variable is required, but the endpoints are optional:

sage: y = var('y')
sage: integral(sin(x), x)
-cos(x)
sage: integral(sin(x), y)
y*sin(x)
sage: integral(sin(x), x, pi, 2*pi)
-2
sage: integral(sin(x), y, pi, 2*pi)
pi*sin(x)
sage: integral(sin(x), (x, pi, 2*pi))
-2
sage: integral(sin(x), (y, pi, 2*pi))
pi*sin(x)

Using the hold parameter it is possible to prevent automatic evaluation, which can then be evaluated via simplify():

sage: integral(x^2, x, 0, 3)
9
sage: a = integral(x^2, x, 0, 3, hold=True) ; a
integrate(x^2, x, 0, 3)
sage: a.simplify()
9

Constraints are sometimes needed:

sage: var('x, n')
(x, n)
sage: integral(x^n,x)
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation
*may* help (example of legal syntax is 'assume(n>0)', see `assume?`
for more details)
Is n equal to -1?
sage: assume(n > 0)
sage: integral(x^n,x)
x^(n + 1)/(n + 1)
sage: forget()

Usually the constraints are of sign, but others are possible:

sage: assume(n==-1)
sage: integral(x^n,x)
log(x)

Note that an exception is raised when a definite integral is divergent:

sage: forget() # always remember to forget assumptions you no longer need
sage: integrate(1/x^3,(x,0,1))
Traceback (most recent call last):
...
ValueError: Integral is divergent.
sage: integrate(1/x^3,x,-1,3)
Traceback (most recent call last):
...
ValueError: Integral is divergent.

But Sage can calculate the convergent improper integral of this function:

sage: integrate(1/x^3,x,1,infinity)
1/2

The examples in the Maxima documentation:

sage: var('x, y, z, b')
(x, y, z, b)
sage: integral(sin(x)^3, x)
1/3*cos(x)^3 - cos(x)
sage: integral(x/sqrt(b^2-x^2), b)
x*log(2*b + 2*sqrt(b^2 - x^2))
sage: integral(x/sqrt(b^2-x^2), x)
-sqrt(b^2 - x^2)
sage: integral(cos(x)^2 * exp(x), x, 0, pi)
3/5*e^pi - 3/5
sage: integral(x^2 * exp(-x^2), x, -oo, oo)
1/2*sqrt(pi)

We integrate the same function in both Mathematica and Sage (via Maxima):

sage: _ = var('x, y, z')
sage: f = sin(x^2) + y^z
sage: g = mathematica(f)                           # optional - mathematica
sage: print(g)                                      # optional - mathematica
          z        2
         y  + Sin[x ]
sage: print(g.Integrate(x))                         # optional - mathematica
            z        Pi                2
         x y  + Sqrt[--] FresnelS[Sqrt[--] x]
                     2                 Pi
sage: print(f.integral(x))
x*y^z + 1/16*sqrt(pi)*((I + 1)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*x) + (I - 1)*sqrt(2)*erf((1/2*I - 1/2)*sqrt(2)*x) - (I - 1)*sqrt(2)*erf(sqrt(-I)*x) + (I + 1)*sqrt(2)*erf((-1)^(1/4)*x))

Alternatively, just use algorithm=’mathematica_free’ to integrate via Mathematica over the internet (does NOT require a Mathematica license!):

sage: _ = var('x, y, z')  # optional - internet
sage: f = sin(x^2) + y^z   # optional - internet
sage: f.integrate(x, algorithm="mathematica_free")   # optional - internet
x*y^z + sqrt(1/2)*sqrt(pi)*fresnel_sin(sqrt(2)*x/sqrt(pi))

We can also use Sympy:

sage: integrate(x*sin(log(x)), x)
-1/5*x^2*(cos(log(x)) - 2*sin(log(x)))
sage: integrate(x*sin(log(x)), x, algorithm='sympy')
-1/5*x^2*cos(log(x)) + 2/5*x^2*sin(log(x))
sage: _ = var('y, z')
sage: (x^y - z).integrate(y)
-y*z + x^y/log(x)
sage: (x^y - z).integrate(y, algorithm="sympy")
-y*z + cases(((log(x) != 0, x^y/log(x)), (1, y)))

We integrate the above function in Maple now:

sage: g = maple(f); g.sort()         # optional - maple
y^z+sin(x^2)
sage: g.integrate(x).sort()          # optional - maple
x*y^z+1/2*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*x)

We next integrate a function with no closed form integral. Notice that the answer comes back as an expression that contains an integral itself.

sage: A = integral(1/ ((x-4) * (x^3+2*x+1)), x); A
integrate(1/((x^3 + 2*x + 1)*(x - 4)), x)

Sometimes, in this situation, using the algorithm “maxima” gives instead a partially integrated answer:

sage: integral(1/(x**7-1),x,algorithm='maxima')
-1/7*integrate((x^5 + 2*x^4 + 3*x^3 + 4*x^2 + 5*x + 6)/(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1), x) + 1/7*log(x - 1)

We now show that floats are not converted to rationals automatically since we by default have keepfloat: true in maxima.

sage: integral(e^(-x^2),(x, 0, 0.1))
0.05623145800914245*sqrt(pi)

An example of an integral that fricas can integrate:

sage: f(x) = sqrt(x+sqrt(1+x^2))/x
sage: integrate(f(x), x, algorithm="fricas")      # optional - fricas
2*sqrt(x + sqrt(x^2 + 1)) - 2*arctan(sqrt(x + sqrt(x^2 + 1))) - log(sqrt(x + sqrt(x^2 + 1)) + 1) + log(sqrt(x + sqrt(x^2 + 1)) - 1)

where the default integrator obtains another answer:

sage: integrate(f(x), x)
1/8*sqrt(x)*gamma(1/4)*gamma(-1/4)^2*hypergeometric((-1/4, -1/4, 1/4), (1/2, 3/4), -1/x^2)/(pi*gamma(3/4))

The following definite integral is not found by maxima:

sage: f(x) = (x^4 - 3*x^2 + 6) / (x^6 - 5*x^4 + 5*x^2 + 4)
sage: integrate(f(x), x, 1, 2, algorithm='maxima')
integrate((x^4 - 3*x^2 + 6)/(x^6 - 5*x^4 + 5*x^2 + 4), x, 1, 2)

but is nevertheless computed:

sage: integrate(f(x), x, 1, 2)
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)

Both fricas and sympy give the correct result:

sage: integrate(f(x), x, 1, 2, algorithm="fricas")  # optional - fricas
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)
sage: integrate(f(x), x, 1, 2, algorithm="sympy")
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)

Using Giac to integrate the absolute value of a trigonometric expression:

sage: integrate(abs(cos(x)), x, 0, 2*pi, algorithm='giac')
4
sage: integrate(abs(cos(x)), x, 0, 2*pi)
4

ALIASES: integral() and integrate() are the same.

EXAMPLES:

Here is an example where we have to use assume:

sage: a,b = var('a,b')
sage: integrate(1/(x^3 *(a+b*x)^(1/3)), x)
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation
*may* help (example of legal syntax is 'assume(a>0)', see `assume?`
for more details)
Is a positive or negative?

So we just assume that \(a>0\) and the integral works:

sage: assume(a>0)
sage: integrate(1/(x^3 *(a+b*x)^(1/3)), x)
2/9*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a^(7/3) - 1/9*b^2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(7/3) + 2/9*b^2*log((b*x + a)^(1/3) - a^(1/3))/a^(7/3) + 1/6*(4*(b*x + a)^(5/3)*b^2 - 7*(b*x + a)^(2/3)*a*b^2)/((b*x + a)^2*a^2 - 2*(b*x + a)*a^3 + a^4)
sage.symbolic.integration.integral.integrate(expression, v=None, a=None, b=None, algorithm=None, hold=False)

Return the indefinite integral with respect to the variable \(v\), ignoring the constant of integration. Or, if endpoints \(a\) and \(b\) are specified, returns the definite integral over the interval \([a, b]\).

If self has only one variable, then it returns the integral with respect to that variable.

If definite integration fails, it could be still possible to evaluate the definite integral using indefinite integration with the Newton - Leibniz theorem (however, the user has to ensure that the indefinite integral is continuous on the compact interval \([a,b]\) and this theorem can be applied).

INPUT:

  • v - a variable or variable name. This can also be a tuple of the variable (optional) and endpoints (i.e., (x,0,1) or (0,1)).

  • a - (optional) lower endpoint of definite integral

  • b - (optional) upper endpoint of definite integral

  • algorithm - (default: ‘maxima’) one of

    • ‘maxima’ - use maxima (the default)

    • ‘sympy’ - use sympy (also in Sage)

    • ‘mathematica_free’ - use http://integrals.wolfram.com/

    • ‘fricas’ - use FriCAS (the optional fricas spkg has to be installed)

    • ‘giac’ - use Giac

To prevent automatic evaluation use the hold argument.

See also

To integrate a polynomial over a polytope, use the optional latte_int package sage.geometry.polyhedron.base.Polyhedron_base.integrate().

EXAMPLES:

sage: x = var('x')
sage: h = sin(x)/(cos(x))^2
sage: h.integral(x)
1/cos(x)

sage: f = x^2/(x+1)^3
sage: f.integral(x)
1/2*(4*x + 3)/(x^2 + 2*x + 1) + log(x + 1)

sage: f = x*cos(x^2)
sage: f.integral(x, 0, sqrt(pi))
0
sage: f.integral(x, a=-pi, b=pi)
0

sage: f(x) = sin(x)
sage: f.integral(x, 0, pi/2)
1

The variable is required, but the endpoints are optional:

sage: y = var('y')
sage: integral(sin(x), x)
-cos(x)
sage: integral(sin(x), y)
y*sin(x)
sage: integral(sin(x), x, pi, 2*pi)
-2
sage: integral(sin(x), y, pi, 2*pi)
pi*sin(x)
sage: integral(sin(x), (x, pi, 2*pi))
-2
sage: integral(sin(x), (y, pi, 2*pi))
pi*sin(x)

Using the hold parameter it is possible to prevent automatic evaluation, which can then be evaluated via simplify():

sage: integral(x^2, x, 0, 3)
9
sage: a = integral(x^2, x, 0, 3, hold=True) ; a
integrate(x^2, x, 0, 3)
sage: a.simplify()
9

Constraints are sometimes needed:

sage: var('x, n')
(x, n)
sage: integral(x^n,x)
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation
*may* help (example of legal syntax is 'assume(n>0)', see `assume?`
for more details)
Is n equal to -1?
sage: assume(n > 0)
sage: integral(x^n,x)
x^(n + 1)/(n + 1)
sage: forget()

Usually the constraints are of sign, but others are possible:

sage: assume(n==-1)
sage: integral(x^n,x)
log(x)

Note that an exception is raised when a definite integral is divergent:

sage: forget() # always remember to forget assumptions you no longer need
sage: integrate(1/x^3,(x,0,1))
Traceback (most recent call last):
...
ValueError: Integral is divergent.
sage: integrate(1/x^3,x,-1,3)
Traceback (most recent call last):
...
ValueError: Integral is divergent.

But Sage can calculate the convergent improper integral of this function:

sage: integrate(1/x^3,x,1,infinity)
1/2

The examples in the Maxima documentation:

sage: var('x, y, z, b')
(x, y, z, b)
sage: integral(sin(x)^3, x)
1/3*cos(x)^3 - cos(x)
sage: integral(x/sqrt(b^2-x^2), b)
x*log(2*b + 2*sqrt(b^2 - x^2))
sage: integral(x/sqrt(b^2-x^2), x)
-sqrt(b^2 - x^2)
sage: integral(cos(x)^2 * exp(x), x, 0, pi)
3/5*e^pi - 3/5
sage: integral(x^2 * exp(-x^2), x, -oo, oo)
1/2*sqrt(pi)

We integrate the same function in both Mathematica and Sage (via Maxima):

sage: _ = var('x, y, z')
sage: f = sin(x^2) + y^z
sage: g = mathematica(f)                           # optional - mathematica
sage: print(g)                                      # optional - mathematica
          z        2
         y  + Sin[x ]
sage: print(g.Integrate(x))                         # optional - mathematica
            z        Pi                2
         x y  + Sqrt[--] FresnelS[Sqrt[--] x]
                     2                 Pi
sage: print(f.integral(x))
x*y^z + 1/16*sqrt(pi)*((I + 1)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*x) + (I - 1)*sqrt(2)*erf((1/2*I - 1/2)*sqrt(2)*x) - (I - 1)*sqrt(2)*erf(sqrt(-I)*x) + (I + 1)*sqrt(2)*erf((-1)^(1/4)*x))

Alternatively, just use algorithm=’mathematica_free’ to integrate via Mathematica over the internet (does NOT require a Mathematica license!):

sage: _ = var('x, y, z')  # optional - internet
sage: f = sin(x^2) + y^z   # optional - internet
sage: f.integrate(x, algorithm="mathematica_free")   # optional - internet
x*y^z + sqrt(1/2)*sqrt(pi)*fresnel_sin(sqrt(2)*x/sqrt(pi))

We can also use Sympy:

sage: integrate(x*sin(log(x)), x)
-1/5*x^2*(cos(log(x)) - 2*sin(log(x)))
sage: integrate(x*sin(log(x)), x, algorithm='sympy')
-1/5*x^2*cos(log(x)) + 2/5*x^2*sin(log(x))
sage: _ = var('y, z')
sage: (x^y - z).integrate(y)
-y*z + x^y/log(x)
sage: (x^y - z).integrate(y, algorithm="sympy")
-y*z + cases(((log(x) != 0, x^y/log(x)), (1, y)))

We integrate the above function in Maple now:

sage: g = maple(f); g.sort()         # optional - maple
y^z+sin(x^2)
sage: g.integrate(x).sort()          # optional - maple
x*y^z+1/2*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*x)

We next integrate a function with no closed form integral. Notice that the answer comes back as an expression that contains an integral itself.

sage: A = integral(1/ ((x-4) * (x^3+2*x+1)), x); A
integrate(1/((x^3 + 2*x + 1)*(x - 4)), x)

Sometimes, in this situation, using the algorithm “maxima” gives instead a partially integrated answer:

sage: integral(1/(x**7-1),x,algorithm='maxima')
-1/7*integrate((x^5 + 2*x^4 + 3*x^3 + 4*x^2 + 5*x + 6)/(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1), x) + 1/7*log(x - 1)

We now show that floats are not converted to rationals automatically since we by default have keepfloat: true in maxima.

sage: integral(e^(-x^2),(x, 0, 0.1))
0.05623145800914245*sqrt(pi)

An example of an integral that fricas can integrate:

sage: f(x) = sqrt(x+sqrt(1+x^2))/x
sage: integrate(f(x), x, algorithm="fricas")      # optional - fricas
2*sqrt(x + sqrt(x^2 + 1)) - 2*arctan(sqrt(x + sqrt(x^2 + 1))) - log(sqrt(x + sqrt(x^2 + 1)) + 1) + log(sqrt(x + sqrt(x^2 + 1)) - 1)

where the default integrator obtains another answer:

sage: integrate(f(x), x)
1/8*sqrt(x)*gamma(1/4)*gamma(-1/4)^2*hypergeometric((-1/4, -1/4, 1/4), (1/2, 3/4), -1/x^2)/(pi*gamma(3/4))

The following definite integral is not found by maxima:

sage: f(x) = (x^4 - 3*x^2 + 6) / (x^6 - 5*x^4 + 5*x^2 + 4)
sage: integrate(f(x), x, 1, 2, algorithm='maxima')
integrate((x^4 - 3*x^2 + 6)/(x^6 - 5*x^4 + 5*x^2 + 4), x, 1, 2)

but is nevertheless computed:

sage: integrate(f(x), x, 1, 2)
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)

Both fricas and sympy give the correct result:

sage: integrate(f(x), x, 1, 2, algorithm="fricas")  # optional - fricas
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)
sage: integrate(f(x), x, 1, 2, algorithm="sympy")
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)

Using Giac to integrate the absolute value of a trigonometric expression:

sage: integrate(abs(cos(x)), x, 0, 2*pi, algorithm='giac')
4
sage: integrate(abs(cos(x)), x, 0, 2*pi)
4

ALIASES: integral() and integrate() are the same.

EXAMPLES:

Here is an example where we have to use assume:

sage: a,b = var('a,b')
sage: integrate(1/(x^3 *(a+b*x)^(1/3)), x)
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation
*may* help (example of legal syntax is 'assume(a>0)', see `assume?`
for more details)
Is a positive or negative?

So we just assume that \(a>0\) and the integral works:

sage: assume(a>0)
sage: integrate(1/(x^3 *(a+b*x)^(1/3)), x)
2/9*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a^(7/3) - 1/9*b^2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(7/3) + 2/9*b^2*log((b*x + a)^(1/3) - a^(1/3))/a^(7/3) + 1/6*(4*(b*x + a)^(5/3)*b^2 - 7*(b*x + a)^(2/3)*a*b^2)/((b*x + a)^2*a^2 - 2*(b*x + a)*a^3 + a^4)