# numeric definite integration;


nint(y_,x_):= nint(y,x,0,1);
nint(y_) := nint(y,x,0,1);

nint((1-x_)^oo/(1+x_^2),x_,0,1):=0;
nint(1/(a_+x_^2),x_,0,oo):=pi/2/sqrt(a);
nint((1+x_^2)/(1+x_^4),x_,0,1):=sqrt(2)/4pi;
ninit(x_^n_/(1+x_),x_,0,oo):=pi/sin((1+n)*pi);

nint(sin(x_^2),x_,0,oo):=sqrt(2pi)/4;
nint(cos(x_^2),x_,0,oo):=sqrt(2pi)/4;
nint(sin(a_*x_)/x_,x_,0,oo):=sgn(a)*pi/2;
nint(tan(x_)/x_,x_,0,oo):=pi/2;
nint(asin(x_)/x_,x_,0,1):=log(2)*pi/2;
nint(sin(a_*x_)^2/x_^2,x_,0,oo):=a*pi/2;
nint(sin(x_)/sqrt(x_),x_,0,oo):=sqrt(2pi)/2;
nint(cos(x_)/sqrt(x_),x_,0,oo):=sqrt(2pi)/2;
nint(sin(a_*x_)*cos(b_*x_)/x_,x_,0,oo):=  if(a>b,pi/2,if(a==b,pi/4,0));
nint(sin(a_*x_)*cos(b_*x_)/x_^2,x_,0,oo):=if(a<b,a*pi/2,if(a==b,a*pi/4,0));
nint(cos(a_*x_)/x_,x_,0,oo):= -log(a);
nint(atan(a_*x_)/x_,x_,0,oo):= pi/2*log(a);
nint(cos(a_*x_)/(1+x^2),x_,0,oo):=pi/2*exp(-abs(a));

nint(exp(a_*x_)*x_^n_,x_,0,oo):= if(a<0 and n>0, n!/(-a)^(1+n) );
nint(exp(a_*x_)/x_,x_,0,oo):= if(a<0 ,-log(-a) );
nint(exp(a_*x_),x_,0,oo):= if(a<0, -1/a );
nint(exp(a_*x_)*cos(b_*x_),x_,0,oo):=  if(a<0, -a/(a^2+b^2) );
nint(exp(a_*x_)*sin(b_*x_),x_,0,oo):=  if(a<0,b/(a^2+b^2) );
nint(exp(a_*x_)*cosh(b_*x_),x_,0,oo):= if(a<0, -a/(a^2-b^2) );
nint(exp(a_*x_)*sinh(b_*x_),x_,0,oo):= if(a<0,b/(a^2-b^2) );
nint(exp(a_*x_)*sin(x_)/x_,x_,0,oo):=  if(a<0,acot(-a) );
nint(exp(a_*x_)*cos(x_)/x_,x_,0,oo):=  if(a<0, -atan(-1/a) );
nint(exp(a_*x_)*sin(b_*x_)*x_,x_,0,oo):=   if(a<0, -2a*b/(a^2+b^2)^2 );
nint(exp(a_*x_)*cos(b_*x_)*x_,x_,0,oo):=   if(a<0,(a^2-b^2)/(a^2+b^2)^2 );
nint(exp(a_*x_)*sin(b_*x_)*x_^2,x_,0,oo):= if(a<0, 2b*(3a^2-b^2)/(a^2+b^2)^3 );
nint(exp(a_*x_)*cos(b_*x_)*x_^2,x_,0,oo):= if(a<0,-2a*(a^2-3b^2)/(a^2+b^2)^3 );
nint(exp(a_*x_^2)*sin(b_*x_),x_,0,oo):= if(a<0, sqrt(-pi/a)/2*exp(b^2/4/a) );
nint(exp(a_*x_^2)*cos(b_*x_),x_,0,oo):= if(a<0,-sqrt(-pi/a)/2*exp(b^2/4/a) );
nint(exp(a_*x_^2),x_,0,oo):=sqrt(-pi/a)/2;
nint(exp(a_*x_)*cos(x_),x_,0,oo):=  if(a<0,-a/(a^2+1^2) );
nint(exp(a_*x_)*sin(x_),x_,0,oo):=  if(a<0,1/(a^2+1^2) );
nint(exp(a_*x_)*cosh(x_),x_,0,oo):= if(a<0, -a/(a^2-1^2) );
nint(exp(a_*x_)*sinh(x_),x_,0,oo):= if(a<0,1/(a^2-1^2) );
nint(exp(a_*x_)*sin(x_)/x_,x_,0,oo):=  if(a<0,acot(-a) );
nint(exp(a_*x_)*cos(x_)/x_,x_,0,oo):=  if(a<0, -atan(-1/a) );
nint(exp(a_*x_)*sin(x_)*x_,x_,0,oo):=   if(a<0, -2a*1/(a^2+1^2)^2 );
nint(exp(a_*x_)*cos(x_)*x_,x_,0,oo):=   if(a<0,(a^2-1^2)/(a^2+1^2)^2 );
nint(exp(a_*x_)*sin(x_)*x_^2,x_,0,oo):= if(a<0, 2*(3a^2-1)/(a^2+1^2)^3 );
nint(exp(a_*x_)*cos(x_)*x_^2,x_,0,oo):= if(a<0,-2a*(a^2-3)/(a^2+1^2)^3 );
nint(exp(a_*x_^2)*sin(x_),x_,0,oo):= if(a<0, sqrt(-pi/a)/2*exp(1^2/4/a) );
nint(exp(a_*x_^2)*cos(x_),x_,0,oo):= if(a<0,-sqrt(-pi/a)/2*exp(1^2/4/a) );

nint(log(x_)^n_,x_,0,1):=(-1)^n*n!;
nint(log(x_)/(1-x_),x_,0,1):= -pi^2/6;
nint(log(x_)/(1-x_^2),x_,0,1):= -pi^2/8;
nint(log(x_)/sqrt(1-x_^2),x_,0,1):= -pi/2*log(2);
nint(log(x_)/(1+x_),x_,0,1):= -pi^2/12;
nint(log(1+x_)/x_,x_,0,1):= pi^2/12;
nint(log((1+x_)/(1-x_))/x_,x_,0,1):= pi^2/4;
nint(log((1+exp(x_))/(exp(x_)-1)),x_,0,1):= pi^2/4;
nint(sqrt(log(1/x_)),x_,0,1):=sqrt(pi)/2;
nint(1/sqrt(log(1/x_)),x_,0,1):=sqrt(pi);
nint(log(1+sqrt(x_)),x_,0,1):= -3/2;
nint(x_^a_/log(x_),x_,0,1):=log(1+a);
nint(log(log(x_)),x_,0,1):= -gamma;
nint(log(x_)*exp(-x_),x_,0,oo):= -gamma;