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6.20.1  Antiderivative and definite integral: integrate int Int

The int and integrate commands compute a primitive or a definite integral. A difference between the two commands is that if you input quest() just after the evaluation of integrate, the answer is written with the ∫ symbol.

Int is the inert form of integrate; namely, it evaluates to integrate for later evaluation.


To find a primitive (an antiderivative):


Examples.


To evaluate a definite integral:


Examples.


Int is the inert form of integrate, it prevents evaluation, for example to avoid a symbolic computation that might not be successful if you just want a numeric evaluation.


Example.
Input:

evalf(Int(exp(x^2),x,0,1))

or:

evalf(int(exp(x^2),x,0,1))

Output:

1.46265174591


Exercises.

  1. Let
    f(x)=
    x
    x2−1
    +ln(
    x+1
    x−1
    )
    Find a primitive of f.
    Input:
    int(x/(x^2-1)+ln((x+1)/(x-1)))
    Output:
    x ln


    x+1
    x−1



    +
    2
    2
     ln
    x2−1
    +
    ln
    x2−1
    2
    Alternatively, define the function f,
    Input:
    f(x):=x/(x^2-1)+ln((x+1)/(x-1))
    then:
    int(f(x))
    The output, of course, will be the same.


    Warning.
    For Xcas, log is the natural logarithm (like ln); log10 is the base-10 logarithm.

  2. Compute:
    2
    x6+2 · x4+x2
      dx
    Input:
    int(2/(x^6+2*x^4+x^2))
    Output:




    −3 x2−2

    x3+x
    3
    2
     arctanx



  3. Compute:
    1
    sin(x)+sin(2 · x )
      dx
    Input:
    integrate(1/(sin(x)+sin(2*x )))
    Output:



    ln


    1−cosx
    1+cosx



    12
    ln


    1−cosx
    1+cosx
    −3


    3




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