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SYMBMATH.H08
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7.6 Integration
SymbMath system itself can find integrals of x^m*e^(x^n),
x^m*e^(-x^n), e^((a*x+b)^n), e^(-(a*x+b)^n), x^m*ln(x)^n, ln(a*x+b)^n,
etc., where m and n are any real number.
The package 'inte.sm' and/or 'd.sm' should be included before
symbolic integration so that it become more powerful on integration. It
is recommended that to expand the integrand by the function expand()
and/or by setting the switch expand=on before symbolic integration.
If symbolic integration fails, the user can define the simple
integral or derivative, then evaluates the integration again (see
7.13 Learning from User).
If the user wants numerical integration by ninte(), the
package 'NInte.sm' should be included before doing numerical
integration (see 8.5 Numeric Integration Package).
7.6.1 Indefinite Integration
SymbMath finds indefinite integrals by functions
inte(expr, x)
Note that the arbitrary constant is not represented.
Example 7.6.1. Find indefinite integrals.
Input:
inte(sinh(x)*e^sinh(x)*cosh(x), x)
inte(sinh(x)^2*cosh(x), x)
inte(x^1.5*exp(x), x)
end
Output:
-e^sinh(x) + sinh(x)*e^sinh(x)
(1/3)*sinh(x)^3
ei(1.5, x)
Example 7.6.2. Find indefinite double integrals.
Input:
inte(inte(x*y, x), y)
end
Output:
(1/4)*x^2*y^2
Example 7.6.3. Find the line integral.
Input:
x=2*t
y=3*t
z=5*t
u=x+y
v=x-y
w=x+y+z
inte((u*d(u,t)+v*d(v,t)+w*d(w,t), t)
end
Output:
63*t^2
Example 7.6.4. Find the integral of sin(x)/x by the mean of
the 'inte.sm' package.
Input:
include('inte.sm')
inte(sin(x)/x, x)
end
Output:
done
si(x)
Defining an integral is similar to defining a rule.
Example 7.6.5
Input:
inte(sin(x_)/x_, x_) := si(x)
inte(sin(t)/t, t)
end
Output:
inte(sin(x_)/x_, x_) := si(x)
si(t)
7.6.2 Definite Integration
SymbMath finds definite integrals by functions
inte(expr, x, xmin, xmax)
Example 7.6.6. Find the definite integral of y=exp(1-x) with
respect to x taken from x=0 to x=infinity.
Input:
inte(exp(1-x), x from 0 to inf)
end
Output:
e
Example 7.6.7. Discontinuous integration of 1/x^2 and 1/x^3
with discontinuity at x=0.
Input:
inte(1/x^2, x from -1 to 2)
inte(1/x^3, x from -1 to 1)
end
Output:
inf
0