Use the Calculus > Integrate command or press Ctrl+Shift+I to find the definite integral or indefinite integral (antiderivative) of an expression.  This command allows you to select the integration variable, and definite or indefinite integration.  If you request definite integration, you can enter the lower and upper limits.  If you request indefinite integration, you can enter the integration constant.  Note that the #n notation can be used to refer to previous expressions when entering integration limits.


Alternatively, the antiderivative of the expression u with respect to the variable x without an integration constant can be entered by typing in an expression of the form INT(u, x).  The antiderivative of the expression u with respect to the variable x with the integration constant c can be entered by typing in an expression of the form INT(u, x, c).  The definite integral of the expression u with respect to the variable x from the lower limit a to the upper limit b can be entered by typing in an expression of the form INT(u, x, a, b).


Antiderivatives and definite integrals are displayed using an integral sign.  Upon simplification, Derive attempts to determine a closed-form expression whose partial derivative is the integrand.  A closed-form expression does not contain infinite series, infinite products, integrals, or limits.  For example,

INT(x/(a^2+b^2·x^4),x)

simplifies to

           2   
        b·x    
 ATAN (——————) 
          a    
———————————————
      2·a·b    


Use the Calculus > Differentiate command to find higher order antiderivatives by entering a negative integer for the order of the derivative.  Alternatively, the nth order antiderivative of the expression u with respect to the variable x can be entered by typing in an expression of the form DIF(u, x, -n).  For example, the second-order antiderivative

DIF(-TAN(x)^2-1, x, -2)

simplifies to

LN(COS(x))


Issue successive Calculus > Integrate commands or enter nested calls on INT to obtain iterated definite integrals.  This provides a convenient method for computing double integrals and triple integrals.  For example, the iterated definite integral

INT(INT(x·LN(y), y, x, 2·x), x, 0, 2)

simplifies to

8·LN(2) - 32/9


Derive uses powerful techniques that enable it to find antiderivatives for a large class of expressions.  These expressions include extended polynomials; products of polynomials with sinusoids, logarithms, or exponentials that are linear in their arguments; piecewise continuous functions; and many rational expressions or expressions involving square roots of linear or quadratic expressions.


If a simplified result includes one or more integral signs it is because the antiderivative does not exist or Derive is unable to find the antiderivative.


If Derive determines that a closed-form antiderivative does not exist in terms of its pre-defined functions and operators, it displays the message No elementary integral on the status line.


If Derive is unable to find an antiderivative that does exist, you may be able to help it along by making a change of variable or using integration by parts.  Functions to facilitate both these integration techniques are defined in the MiscellaneousFunctions.mth144O15P utility file.


If all else fails, you can approximate the antiderivative of an expression by integrating its Taylor polynomial approximation (see Calculus > Taylor Series commandMRJ3J4).  You may need to use several different expansion points to make the series converge more rapidly over the interval of interest.  For example, to find an approximation for the antiderivative of SIN(x)/x by integrating its fourth order Taylor polynomial approximation about the point x=0, simplify the expression

INT(TAYLOR(SIN(x)/x,x,0,4),x)


If you chose to add an arbitrary constant to antiderivatives, use a new constant name each time to prevent unjustified cancellations of the constants when subtracting antiderivatives.


When comparing a Derive produced antiderivative with one in an integral table, it is important to realize that they may appear quite different.  They may be different forms of equivalent expressions, or they may differ by a well disguised constant.  Note that if such different forms occur, you can use Derive to simplify the difference of the two antiderivatives and see if the result is independent of the integration variable.


If precision is in approximate mode and the integrand depends only on the integration variable and both limits are numeric, then Derive uses an extrapolated adaptive Simpson's rule to numerically approximate definite integrals.  This algorithm is also used if precision is in mixed mode and Derive is unable to determine a closed-form antiderivative for the integrand.


The computing time for this numerical integration algorithm usually increases sharply with the current precision.  For many integrals the resulting accuracy is nearly that of the current precision.  However, the accuracy may be significantly worse than the current precision if the integrand or one of its low order derivatives has a discontinuity or a singularity within the interval or at either endpoint.  Integrals that have relatively equal areas above and below the axis can also cause poor relative error via catastrophic cancellation.  Derive is usually able to recognize that one of these situations has occurred, in which case it displays the warning message Dubious accuracy.


If precision is in exact or mixed mode, Derive simplifies definite integrals by trying to determine a closed-form antiderivative for the integrand.  Then it subtracts the limit of the antiderivative as the integration variable approaches the lower limit from the limit as the integration variable approaches the upper limit.  Both limits are approached from within the interval.  Thus, Derive can determine some improper integrals having infinite limits or integrand singularities at the limits.  For example,

INT(1/x^2,x,1,inf)

simplifies to 1, and

INT(1/SQRT(x),x,0,1)

simplifies to 2.


Although this technique is valid for endpoint singularities, it may be invalid if there is a singularity within the integration interval.  Instead of the Riemann integral, it may yield the Cauchy principal value.  For example,

INT(1/x^3,x,-1,2)

simplifies to 3/8 which is correct only if the assumption is made that the singularity at 0 is approached at the same rate from both sides, thus exactly canceling the positive and negative infinite areas.


An even more serious problem occurs when a difference of antiderivatives across a singularity gives results that even Cauchy would call wrong.  As is the case when using integral tables, it is your responsibility to find singularities that occur within the interval of integration and to split the integrals there.  


If the integrand looks as though it may have singularities but their locations are not obvious, try using the soLve command to find the zeros of the integrand's common denominator.  Plotting the integrand over the integration interval may also help.  However, there is no known method that is guaranteed to find all internal singularities.


There may exist a closed-form definite integral of an expression for which there is no closed-form antiderivative.  Although Derive can not directly integrate such expressions, it is useful for doing the steps required by various advanced techniques required to solve such problems.


Many applications of integration, including functions for Laplace transforms, Fourier series, arc lengths, surface areas, centroids, and moments of inertia, are defined in the IntegrationApplications.mth2W1Y_YU utility file.


Other Calculus commandsCalculus_commands 

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