Numerical integration is discussed in §2.2.2 below.
Sage can integrate some simple functions on its own:
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
Sage can also compute symbolic definite integrals involving limits.
sage: var('x, k, w') (x, k, w) sage: f = x^3 * e^(k*x) * sin(w*x) sage: print f.integrate(x) 6 3 4 5 2 7 3 (((k w + 3 k w + 3 k w + k ) x 6 2 4 4 2 6 2 4 3 2 5 + (3 w + 3 k w - 3 k w - 3 k ) x + (- 18 k w - 12 k w + 6 k ) x 4 2 2 4 k x - 6 w + 36 k w - 6 k ) e sin(w x) 7 2 5 4 3 6 3 5 3 3 5 2 + ((- w - 3 k w - 3 k w - k w) x + (6 k w + 12 k w + 6 k w) x 5 2 3 4 3 3 k x + (6 w - 12 k w - 18 k w) x - 24 k w + 24 k w) e cos(w x)) 8 2 6 4 4 6 2 8 /(w + 4 k w + 6 k w + 4 k w + k ) sage: integrate(1/x^2, x, 1, infinity) 1