(*Video 1: Integration by substitution *)

(* Examples *)
HoldForm[u=g[x]]
HoldForm[Integrate[f[g[x]]g'[x],x]]==Integrate[f[u],u]

(*Example 1*)

HoldForm[Integrate[20 (x^3) Sqrt[5x^4+7],x]]

g[x_]:=7+5 (x^4)
f[u_]:=Sqrt[u]
g'[x]f[g[x]]

Integrate[f[u],u]/.u->g[x]

Integrate[20 (x^3) Sqrt[5x^4+7],x]

(*Example 2*)

HoldForm[Integrate[(10 x+3 (x^2))E^(x^3+5x^2+1),x]]

Integrate[(10 x+3 (x^2))E^(x^3+5x^2+1),x]

g[x_]:=(x^3)+5 (x^2)+1
f[u_]:=E^u

Integrate[E^u,u]/.u->g[x]

D[(E^((x^3)+5 (x^2)+1)),x]

(*Example 3*)

HoldForm[Integrate[x(x^2+1)^1500,x]]

Integrate[x(x^2+1)^1500,x]

g[x_]:=(x^2)+1
f[u_]:=u^1500

Integrate[1/2f[u],u]/.u->g[x]

D[(((((x^2)+1))^1501)/3002),x]

(*Video 2: Integration by substitution *)

(*Example 4*)

HoldForm[Integrate[E^(2x) Cos[E^(2x)],x]]

D[E^(2x),x]

g[x_]:=E^(2x)
f[u_]:=Cos[u]/2
Integrate[f[u],u]/.u->g[x]

Integrate[E^(2x) Cos[E^(2x)],x]

(*Example 5*)

HoldForm[Integrate[(Cos[x]^3+2Cos[x]-1)Sin[x],x]]

g[x_]:=Cos[x]
f[u_]:=-(u^3+2u-1)
Integrate[f[u],u]/.u->g[x]

Integrate[(Cos[x]^3+2Cos[x]-1)Sin[x],x]

(*Example 6*)

HoldForm[Integrate[1/(x Log[x]),x]]

g[x_]:=Log[x];
f[u_]:=1/u;
Integrate[f[u],u]/.u->g[x]

Integrate[1/(x Log[x]),x]

D[Log[Log[x]],x]