restart;
#Q1
I1:=Int((1+u)^(1/3),u=1..2);
evalf(I1);
I2:=Int((cos(x)+sin(x)^2)/(sin(x)+cos(x)^2),x=0..Pi);
evalf(I2);
I3:=Int(1/sqrt(1+y+y^2+y^3),y=0..1);
evalf(I3);
I4:=Int(ln(t)^8,t=1..exp(1));
evalf(I4);
#Q2
B:=5^(2*n+2)/(2*n+2)!;
0.5*10^(-4);
evalf(subs(n=8,B));
evalf(subs(n=9,B));
#Q3
f:=2*cos(t)+cos(5*t);
g:=3*sin(t);
df:=diff(f,t);
dg:=diff(g,t);
ddf:=diff(df,t);
ddg:=diff(dg,t);
L:=Int(sqrt(df^2+dg^2),t=0..2*Pi);
evalf(L);
A:=Int(dg*f,t=0..2*Pi);
evalf(A);
t0:=arcsin(sqrt(2)/2);
m:=subs(t=t0,dg/df);
simplify(m);
kappa:=(ddg*df-dg*ddf)/(df^2+dg^2)^(3/2);
simplify(subs(t=t0,kappa));
#Q6
f:=4*sin(x);
g:=sqrt(6)-sqrt(2);
a:=solve(f=g,x);
evalf(a-Pi/12);
# The exact value of a
a:=Pi/12;
b:=Pi-a;
Vx:=Int(Pi*(f^2-g^2),x=a..b);
Vx2:=value(Vx);
trig1:=sin(Pi/12)=sqrt((1-cos(Pi/6))/2);
trig2:=cos(Pi/12)=sqrt((1+cos(Pi/6))/2);
Vx3:=subs([trig1,trig2],Vx2);
simplify(Vx3);
evalf(Vx3);
Vy:=Int(2*Pi*x*(f-g),x=a..b);
Vy2:=value(Vy);
Vy3:=subs([trig1,trig2],Vy2);
factor(Vy3);
evalf(Vy3);
evalf(sqrt(2)/6*Pi^2*(5*Pi*(1-sqrt(3))+12*(1+sqrt(3))));
#Q7
limit((sin(x)-x*cos(x))/x^3,x=0);