Integral Calculus

Besides the examples on this page, please see the discussion in ["BasicCalculus"].

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: 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

Applications to Areas, Volume, and Averages

Using Integration to Determine Areas Between Curves

Computing Volumes of Surfaces of Revolution

Average Values

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

Integration by parts

Trigonometric integrals

• o Some Remarks on Using Complex-Valued Functions

Trigonometric substitutions

Factoring polynomials

Integration of Rational Functions Using Partial Fractions

Approximating Integrals

Improper Integrals

• o Convergence, Divergence, and Comparison