Antiderivative and definite integral : integrate int Int

### 5.20.1 Antiderivative and definite integral : integrate int Int

integrate (or int) computes 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.

integrate (or int or Int) takes one, two or four arguments.

- with one or two arguments
  an expression or an expression and the name of a variable (by default x),
  integrate (or int) returns a primitive of the expression with respect to the variable given as second argument.
  Input :
  integrate(x`^`2)
  Output :
  x`^`3/3
  Input :
  integrate(t`^`2,t)
  Output :
  t`^`3/3
- with four arguments :
  an expression, a name of a variable and the bounds of the definite integral,
  integrate (or int) returns the exact value of the definite integral if the computation was successful or an unevaluated integral otherwise.
  Input :
  integrate(x`^`2,x,1,2)
  Output :
  7/3
  Input :
  integrate(1/(sin(x)+2),x,0,2*pi)
  Output after simplification (with the simplify command) :
  2*pi*sqrt(3)/3

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.
Input :

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

or :

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

Output :

1.46265174591

Exercise 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*log((x+1)/(x-1))+log(x`^`2-1)+1/2*log(2*x`^`2/2-1)

Or define the function f, input :

f(x):=x/(x`^`2-1)+ln((x+1)/(x-1))

then input :

int(f(x))

Output of course the same result.
Warning
For Xcas, log is the natural logarithm (like ln), as log10 is 10-basis logarithm

Exercise 2
Compute :

∫ |
2
x6+2 · x4+x2
| dx

Input :

int(2/(x`^`6+2*x`^`4+x`^`2))

Output :

2*((3*x`^`2+2)/(-(2*(x`^`3+x)))+-3/2*atan(x))

Exercise 3
Compute :

∫ |
1
sin(x)+sin(2 · x )
| dx

Input :

integrate(1/(sin(x)+sin(2*x )))

Output :

(1/-3*log((tan(x/2))`^`2-3)+1/12*log((tan(x/2))`^`2))*2