Integrate[Abs[(x-1)(x-3)],{x,0,4}]

Integrate[Abs[2x-6],{x,0,4}]

Plot[Abs[2x-6],{x,0,4}]

Plot[Abs[(x-1)(x-3)],{x,0,4}]

Plot[{1+x,2-x},{x,0,1},PlotRange->{0,2}]

a=1;b=2;c=3;d=4;
{(a c+b d)^2,(a^2+c^2)(b^2+d^2)}

a=2;b=5;c=3;d=8;
{(a b+c d)^2,(a^2+c^2)(b^2+d^2)}

a=2;b=5;c=-3;d=8;
{(a b+c d)^2,(a^2+c^2)(b^2+d^2)}

a=-2;b=-51;c=13;d=-8;
{(a b+c d)^2,(a^2+c^2)(b^2+d^2)}

Plot[Floor[1/2x+1],{x,1,5},PlotRange->{0,4},Filling->Axis]

Integrate[Floor[1/2x+1],{x,1,5}]

Solve[k/(1+x)==1/(1+k x),k]

f[x_]:=1/(x^2+2x+2);
Simplify[{f[x],1/2(f[x]+f[-x]),1/2(f[x]-f[-x])}]

f[x]

Plot[f[x],{x,-4,4}]

Simplify[1/2(f[x]+f[-x])]

Simplify[1/2(f[x]-f[-x])]

Expand[1+(((1+x))^2)]

Plot[{(1/(1-(x^2))),(x/(1-(x^2))),(x/(1-(x^2)))+(1/(1-(x^2)))},{x,-4,4}]

Simplify[(x/(1-(x^2)))+(1/(1-(x^2)))]

NIntegrate[f[x],{x,-2,2}]

NIntegrate[1/2(f[x]+f[-x]),{x,-2,2}]

g[x_]:=2x+x^2

h[x_]:=5/x

f[g[1]+h[1]]

f[g[1]]+f[h[1]]

g[f[1]]+h[f[1]]

g[x]+h[x]

(g[x]+h[x])/.x->f[1]

points=Table[{1/n,1/2(1+(-1)^n)},{n,1,30}]

ListLinePlot[points,PlotRange->All,PlotStyle->Thin,Ticks->{Table[1/n,{n,1,7}],{0,1}}]

1/2(1-1/3)+1/2(1/3-1/5)+1/2(1/5-1/7)+1/2(1/7-1/9)

1/2Sum[1/n-1/(n+2),{n,1,Infinity,2}]

2^2-10

Table[{j^2-i},{i,-10,10},{j,1,10}]

Flatten[%]

r[x_]:=If[0<=Abs[(x-1)(x-2)]≤6,1,0];
Plot[r[x],{x,-7,7}]
Integrate[r[x],{x,-5,5}]

Table[Factor[(x-1)(x-2)+i],{i,-6,25}]

Factor[(x-1)(x-2)

r[x_]:=If[3<=Abs[x^2-4]≤5,1,0];
Plot[r[x],{x,-7,7}]
Integrate[r[x],{x,-5,5}]

Solve[Abs[x^2-4]==3,x]
Solve[Abs[x^2-4]==5,x]

2((3-Sqrt[7])+1)

Simplify[%]

Solve[Abs[x^2-4]==5,x]

Plot[Abs[x^2-4],{x,-4,4}]

Solve[Abs[x^2-2]==3]

Table[Abs[x^2-4],{x,-7,7}]

p[x_]:=If[0≤x≤3,1/9x^3+2,If[3≤x≤5,1/9x^2+4,If[5≤x≤6, 8,(x-6)^2+8]]];
Plot[p[x],{x,0,8},Exclusions->{3,5,6},Ticks->{{1,2,3,4,5,6,7,8},{2,4,6,8,10,12}},PlotRange->{0,13}]

Reduce[(8/n)(12-2)<1/10,n]

16.5*2