Calculus.

Calculus
Mathematicaresultexample
Limit[f[x],x->x0]calculate the limit as xx0Limit[(E^x-1)/x,x->0]
Limit[f[x],x->Infinity]calculate the limit as x→∞Limit[1/x,x->Infinity]
Dt[f[x],x]calculates the first derivative of f(x) with respect to xDt[Sin[2x]^2,x]
Dt[f[x]]calculates the total differential of f(x)Dt[Sin[2x]^2]
Dt[f[x],{x,n}]calculates the n-th derivative of f(x) with respect to xDt[Sin[2x]^2,{x,2}]
D[f[x,y],x]calculate the partial derivative of f(x,y) with respect to xD[Sin[x y],x]
D[f[x,y],{x,n}]calculates the n-th partial derivative of f(x,y) with respect to xD[Sin[x y],{x,2}]
Integrate[f[x],x]calculates the indefinite integral of f(x) with respect to xIntegrate[Log[x],x]
Integrate[f[x],{x,xmin,xmax}]calculates the definite integral of f(x) with respect to x in the range [xmin,xmax] Integrate[Log[x],{x,1,E}]
DSolve[equation,y,x]solves a differential equation for the function y, with independent variable xDSolve[y'[x]==y,y,x]
Series[f[x],{x,0,n}]generates n+1 terms of the expansion in MacLaurin power series for f(x)Series[Exp[x],{x,0,10}]
Series[f[x],{x,x0,n}] generates n+1 terms of a Taylor power series expansion for f(x) about the point x0Series[Log[x],{x,1,10}]
Sum[f[i],{i,imin,imax}]calculates the sum of the addends f(i), with i from imin to imax, incrementing i by 1 at each stepSum[2i+1,{i,0,10}]
Sum[f[i],{i,imin,imax,inc}]calculates the sum of the addends f(i), with i from imin to imax, incrementing i by inc at each stepSum[2i+1,{i,0,10}]

Examples with Wolframalpha.

Limit: Limit[(E^x-1)/x,x->0]

Derivative: Dt[Sin[2x]^2,x]

Indefinite integral: Integrate[Log[x],x]

If you click on the button `Show steps`, you will get the demonstration.

Definite integral: Integrate[Log[x],{x,1,E}]

Differential equazione: DSolve[y'[x]==y,y,x]

If you click on the button `Show steps`, you will get the demonstration.

Maclaurin series expansion: Series[Exp[x],{x,0,10}]

The sum of the first 11 odd numbers: Sum[2i+1,{i,0,10}]