(*Video 1: Antiderivatives / Indefinite integral *)

F[x_]:=x^3

F'[x]

Integrate[3x^2,x](*Command to find an antiderivative*)

(*Are antiderivatives unique? 
No! All antiderivatives of the same function differ by constants*)

D[x^3+c ,x]

(*Antiderivative of a polynomial:*)

Integrate[x^n,x] (* n != -1*)

D[((x^(1+n))/(1+n)),x]

Integrate[4x^5+2x^3+1/2 x+1,x]

Integrate[6x^6+4x^8+1/3x^2+12,x]

D[12 x+((x^3)/9)+((6 (x^7))/7)+((4 (x^9))/9)+15,x]

Integrate[x^2,y](*Careful: second argument given is the variable of integration!*)

D[(x^2) y,y]

Integrate[a x^3+b x^2+c x+d,x]

D[d x+((c (x^2))/2)+((b (x^3))/3)+((a (x^4))/4),x]

(*Video 2: Definite integrals*)

HoldForm[Integrate[x^2,x]]==Integrate[x^2,x]
(*Antiderivative of x^2, also known as indefinite integral of x^2 *)

HoldForm[Integrate[x^2,{x,a,b}]](*Definite integral of x^2, from x=a to x=b*)

(*The above definite integral, by definition, is the area under the graph of y=x^2 that lies between x=a and x=b*)

Manipulate[Show[Plot[x^2,{x,0,5}],Plot[x^2,{x,a,b},Filling->Bottom,PlotRange->{{0,5},{0,25}}]],{a,0.01,5},{b,0.02,5}]

Integrate[x^2,{x,2,4}]