# Integral Calculus

## Definite and Indefinite Integrals

SAGE can compute both definite integrals like \int_0^1 \frac{dx}{x^3+1} and indefinite integrals such as \int \frac{dx}{x^3+1}:

sage: print integrate(1/(x^3+1),x)
2 x - 1
2            atan(-------)
log(x  - x + 1)        sqrt(3)    log(x + 1)
- --------------- + ------------- + ----------
6             sqrt(3)          3
sage: integrate(1/(x^3+1), x, 0, 1)
(6*log(2) + sqrt(3)*pi)/18 + sqrt(3)*pi/18

More examples:

sage: integrate(1/x^2, x, 1, infinity)
1
sage: f = x^3
sage: f.integral()
x^4/4
sage: integral(x^3,x)
x^4/4
sage: f = x*sin(x^2)
sage: integral(f,x)
-cos(x^2)/2

### The Definite Integral

• o The definition of area under curve o Relation between velocity and area o Definition of Integral o The Fundamental Theorem of Calculus

## Indefinite Integrals and Change

• o Indefinite Integrals o Physical Intuition

## Substitution and Symmetry

• o The Substitution Rule o The Substitution Rule for Definite Integrals o Symmetry

## Polar coordinates and complex numbers

• o Polar Coordinates o Areas in Polar Coordinates o Complex Numbers o Polar Form o Complex Exponentials and Trig Identities o Trigonometry and Complex Exponentials

## Integration Techniques

### Trigonometric integrals

• o Some Remarks on Using Complex-Valued Functions

### Approximating Integrals

Regarding numerical approximation of \int_a^bf(x) dx , where f is a piecewise defined function, Sage can

• compute (for plotting purposes) the piecewise linear function defined by the trapezoid rule for numerical integration based on a subdivision into N subintervals
• the approximation given by the trapezoid rule,
• compute (for plotting purposes) the piecewise constant function defined by the Riemann sums (left-hand, right-hand, or midpoint) in numerical integration based on a subdivision into N subintervals,
• the approximation given by the Riemann sum approximation.

sage: f1 = x^2
sage: f2 = 5-x^2
sage: f = Piecewise([[(0,1),f1],[(1,2),f2]])
sage: f.trapezoid(4)
Piecewise defined function with 4 parts, [[(0, 1/2), x/2], [(1/2, 1), 9*(x - 1/2)/2 + 1/4], [(1, 3/2), (x - 1)/2 + 5/2], [(3/2, 2), 11/4 - 7*(x - 3/2)/2]]
sage: f.riemann_sum_integral_approximation(6,mode="right")
19/6
sage: f.integral()
3
sage: n(f.integral())
3.00000000000000

### Improper Integrals

• o Convergence, Divergence, and Comparison

Integral_Calculus (last edited 2008-11-14 13:42:08 by localhost)