complex contour integral ...plz help me

I write the limits of integration from 0 to 2*pi when he give me a Full Circle ,, but what I write

instead of 2*pi when he give me half circle as show in figure? :

this statement of matlab when he give me a contour for full circle :

q1=quad(@(t) fun(g(t)).*gprime(t),0,2*pi)

I want instead it by contour for half circle ,, what I write instead( 0,2*pi)?

this problem that I want answer it:

find the integral around half contour

∫(x^3+3)/((x^2+1)(x^2+4))

Answer 1

[Torsten]

fun=@(z)(x.^3+3)./((x.^2+1).*(x.^2+4));
g1=@(theta)R*cos(theta)+1i*R*sin(theta);
g1prime=@(theta)-R*sin(theta)+1i*R*cos(theta);
g2=@(t)-R+t*2*R;
g2prime=2*R;
q=quad(@(t)fun(g1(t)).*gprime(t),0,pi)+quad(@(t)fun(g2(t))*g2prime,0,1);

Best wishes

Torsten.

[Siwei Wang]

I correct a few typos in Torsten's answer:

R=10;

fun=@(x)(x.^3+3)./((x.^2+1).*(x.^2+4));

g1=@(theta)R*cos(theta)+1i*R*sin(theta);

g1prime=@(theta)-R*sin(theta)+1i*R*cos(theta);

g2=@(t)-R+t*2*R;

g2prime=2*R;

q=quad(@(t)fun(g1(t)).*g1prime(t),0,pi)+quad(@(t)fun(g2(t))*g2prime,0,1)

The result is q=1.57079596718350 + 3.14159265928277i