### 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(x^2)

x3
3

integrate(t^2,t)

t3
3

integrate(x^2,x,1,2)

7
3

integrate(1/(sin(x)+2),x,0,2*pi)

2
3

π

√

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, like for example:

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

or:

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

1.46265174591 |  |  |  |  |  |  |  |  |  |

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

⎛
⎜
⎜
⎝

x+1
x−1

⎞
⎟
⎟
⎠

+

2
2

ln

⎪
⎪

x2−1

⎪
⎪

+

ln | ⎪ ⎪ | x2−1 | ⎪ ⎪

2

Compute ∫2/x6+2x4+x2 d x.

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

2

⎛
⎜
⎜
⎜
⎝

−3 x2−2

2 | ⎛ ⎝ | x3+x | ⎞ ⎠

−

3
2

arctan x

⎞
⎟
⎟
⎟
⎠

Compute ∫1/sin(x)+sin(2 x)d x.

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

2

⎛
⎜
⎜
⎝

ln

⎛
⎜
⎜
⎝

1−cos x
1+cos x

⎞
⎟
⎟
⎠

12

−

ln

⎪
⎪
⎪
⎪

1−cos x
1+cos x

−3

⎪
⎪
⎪
⎪

3

⎞
⎟
⎟
⎠