restart;
int(x^n, x);

A := diff(x^n, x);

A;

simplify(A);

F := 1/(2*x^2-4*x-8);

PF := convert(F, parfrac, sqrt(5));

P := 1/F;

solve(P = 0, x);

f := int(PF, x);
lnabs:=x->ln(abs(x));
fabs:=subs(ln=lnabs,ftmp);



g := int(F, x);

f;

g;

Q := simplify(diff(f, x));

simplify(diff(g, x));

1/expand(1/Q, x);

plot([fabs,g], x = -1 .. 2);

F1 := 1/((x-1)*(x-2)*(x-3)^2);

int(F1, x);

convert(F1, parfrac);

int(x^3*exp(x), x);

p := (a+b*x+c*x^2+d*x^3)*exp(x);

eq1 := diff(p, x) = x^3*exp(x);

Y1 := collect(simplify(eq1/exp(x)), x);

e1 := subs(x = 0, lhs(Y1)) = subs(x = 0, rhs(Y1));
e2 := coeff(lhs(Y1), x) = coeff(rhs(Y1), x);
e3 := coeff(lhs(Y1), x^2) = coeff(rhs(Y1), x^2);
e4 := coeff(lhs(Y1), x^3) = coeff(rhs(Y1), x^3);




h := solve({e1, e2, e3, e4}, {a, b, c, d});

subs(h, p);

h1 := arctan(x);

series(h1, x, 10);

t4 := series(3*tan(4*x), x);

t12 := series(12*tan(x), x); 

s4 := series(3*sin(4*x), x); 

s12 := series(12*sin(x), x); 

subs(x = 0, simplify(convert(t4-t12, polynom)/convert(s4-s12, polynom))); 

limit((3*tan(4*x)-12*tan(x))/(3*sin(4*x)-12*sin(x)), x = 0); 

num := 3*tan(4*x)-12*tan(x);
den := 3*sin(4*x)-12*sin(x); 


dnum := diff(num, x$3);
dend := diff(den, x$3);


p := simplify(subs(x = 0, dnum)); q := simplify(subs(x = 0, dend));


p/q;

int(x^2*exp(x^2), x);

2*(int(exp(-t^2), t = 0 .. x))/sqrt(Pi);