     suivant: Discrete summation: sum monter: Integration précédent: Integration   Table des matières   Index

## Antiderivative and definite integral : integrate int Int

integrate (or int) compute the 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 successfull 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 successfull 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) = + ln( )

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 :  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 :  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     suivant: Discrete summation: sum monter: Integration précédent: Integration   Table des matières   Index
giac documentation written by Renée De Graeve and Bernard Parisse