 
 
 
13.3.1  Antiderivatives and definite integrals
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.
- 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
| integrate(1/(sin(x)+2),x,0,2*pi) | 
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, like for example:
| evalf(Int(exp(x^2),x,0,1)) | 
or:
| evalf(int(exp(x^2),x,0,1)) | 
Let  f(x)=x/x2−1+ln(x+1/x−1).
Find a primitive of f.
| int(x/(x^2-1)+ln((x+1)/(x-1))) | 
or:
| f(x):=x/(x^2-1)+ln((x+1)/(x-1)):;
 int(f(x)) | 
|  | | | x ln | ⎛ ⎜
 ⎜
 ⎝
 |  | ⎞ ⎟
 ⎟
 ⎠
 | + |  | ln | ⎪ ⎪
 | x2−1 | ⎪ ⎪
 | + |  | 
 |  |  |  |  |  |  |  |  |  |  | 
 | 
Compute  ∫2/x6+2x4+x2  dx.
|  | | | 2 | ⎛ ⎜
 ⎜
 ⎜
 ⎝
 |  | − |  | arctanx | ⎞ ⎟
 ⎟
 ⎟
 ⎠
 | 
 |  |  |  |  |  |  |  |  |  |  | 
 | 
Compute ∫1/sin(x)+sin(2 x)  dx.
| integrate(1/(sin(x)+sin(2*x))) | 
 
 
