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restart;
#Newton's Methods
f:=x->x+log(x);
plot(f(x),x=0.1..1);
df:=D(f);
phi:=a->a-f(a)/df(a);
a1:=phi(0.5);
a2:=phi(a1);
a3:=phi(a2);
a4:=phi(a3);
I1:=Int(sin(theta),theta=0..Pi/4);
#Changing Variables in an Integral
?Change
with(IntegrationTools);
Change(I1,u=sin(theta));
I2:=Int(log(x),x=1..2);
Change(I2,x=sqrt(t));
I3:=Int(sqrt(1+1/x),x);
A3:=value(I3);
simplify(A3);
Change(I3,t=1/x);
A4:=Change(I3,x=tan(theta));
simplify(A4);
I4:=Int(sqrt(1-1/x),x);
value(I4);
A5:=Change(I4,x=-u);
simplify(A5);
#Defining a function by an Integral
f:=t->cos(t^2);
plot(f(t),t=0..6);
F:=x->Int(f(t),t=0..x);
value(F(0));
G:=x->value(F(x));
G(0);
G(1);
?FresnelC
G(y);
plot(G(y),y=0..6);
plot(G(y),y=0..60,numpoints=1000);
limit(G(y),y=infinity);

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