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):
-
int (or integrate) takes one mandatory
argument and one optional argument:
-
expr, an expression.
- Optionally, x, the name of a
variable (by default the value is x, so if the variable is
x the second argument is unnecessary).
- int(expr ⟨,x⟩)
(or integrate(expr ⟨,x⟩)) returns a
primitive of expr with respect to x.
Examples.
-
Input:
integrate(x^2)
Output:
- Input:
integrate(t^2,t)
Output:
To evaluate a definite integral:
-
int (or integrate)
takes four arguments:
-
expr, an expression.
- x, the variable.
- a and b, the bounds of the definite integral.
- int(expr,x,a,b)
(or integrate(expr,x,a,b)) returns
the exact value of the definite integral if the computation was
successful or an unevaluated integral otherwise.
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:
Exercises.
-
Let
Find a primitive of f.
Input:
int(x/(x^2-1)+ln((x+1)/(x-1)))
Output:
x ln | ⎛
⎜
⎜
⎝ | | ⎞
⎟
⎟
⎠ | + | | ln | ⎪
⎪ | x2−1 | ⎪
⎪ | + | |
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.
- Compute:
Input:
int(2/(x^6+2*x^4+x^2))
Output:
2 | ⎛
⎜
⎜
⎜
⎝ | | − | | arctanx | ⎞
⎟
⎟
⎟
⎠ |
- Compute:
Input:
integrate(1/(sin(x)+sin(2*x )))
Output: