x[a_,t_]:=t
y[a_,t_]:=(a/2)(Exp[t/a]+Exp[-t/a])
Manipulate[ParametricPlot[{x[a,t],y[a,t]},{t,-1,1},PlotRange->{{-1,1},{-1,3}}],{a,1/2,2}]

Integrate[Sqrt[1+(D[Cosh[x],x])^2],{x,-1,1}]

%//N

Show[{ParametricPlot[{x[1,t],y[1,t]},{t,-1,1}],ParametricPlot[{ArcSinh[s],Sqrt[1+s^2]},{s,-1,1},PlotStyle->Thickness[.01]]}]

Integrate[2Cos[t]/Sqrt[2-2Sin[t]],t]

F[t_]:=-2 (Sqrt[2-2 Sin[t]])

F[Pi/2]-F[0]
-(F[3Pi/2]-F[Pi/2])
F[2Pi]-F[3Pi/2]

%+%%+%%%

Integrate[Sqrt[4Cos[t]^2]/Sqrt[2-2Sin[t]],{t,0,2Pi}]

F[Pi/2]

F[Pi/2]-F[0]

PolarPlot[3Sin[θ],{θ,0,5Pi/6},PlotRange->{{-1.5,1.5},{0,3}}]

PolarPlot[1+Sin[θ],{θ,0,2Pi}]

Integrate[Abs[2Cos[θ]]/Sqrt[2-2Sin[θ]],{θ,0,2Pi}]

F[θ_]:=-2Sqrt[2-2Sin[θ]]

(F[Pi/2]-F[0])-(F[3Pi/2]-F[Pi/2])+(F[2Pi]-F[3Pi/2])

(F[Pi/2]-F[0])
-(F[3Pi/2]-F[Pi/2])
(F[2Pi]-F[3Pi/2])

Show[{PolarPlot[1+Sin[θ],{θ,0,Pi/2},PlotStyle->Red],PolarPlot[1+Sin[θ],{θ,Pi/2,3Pi/2}],PolarPlot[1+Sin[θ],{θ,3Pi/2,2Pi},PlotStyle->Yellow]}]

Plot[{Sinh[x],ArcSinh[x]},{x,-10,10},PlotRange->{{-10,10},{-10,10}}]

Plot[Cosh[x],{x,-10,10}]