Basic single-variable integration.

Integrate[ x^2, x ]

Integrate[ x^2, {x,0,3} ]

Integrate multivariable functions similarly to single-variable ones, except you can give multiple variables and bounds. Mathematica takes the first variable and bounds to be the outermost integral, the next to be the second-from-outermost, etc. Notice that the bounds on inner integrals can be expressions involving variables from outer ones.

Integrate[ x*y^2, {x,0,1}, {y,0,1-x} ]

Integrate[ x * y, {x,0,1}, {y, x^2, 1} ]

Mathematica can even handle indefinite multivariable integrals, although exactly what an indefinite multivariable integral should be with regard to the constants of integration is dicey.

Integrate[ x*y^2, x, y ]