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

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:

or:

Let f(x)=x/x²−1+ln(x+1/x−1). Find a primitive of f.

or:

Compute ∫2/x⁶+2x⁴+x²  dx.

Compute ∫1/sin(x)+sin(2 x)  dx.