<s>[INST] evaluate if integral converge & determine antiderivative The problem is i need to 
study the convergence of A and B
and find the  antiderivative  of C
 $$A=\int_0^\infty \frac{\sin(x) +x}{\sqrt x + x^3}dx$$ $$B=\int_0^\infty \frac{1}{\sqrt {e^x-1}(x^2+x^{1/3}  )}dx$$
$$C=\int\frac{1}{x}\sqrt\frac{2-x}{2+x}dx$$
in C i try substitution  $$U =\sqrt\frac{2-x}{2+x}$$
 [/INST] hi sami i look if the problem C  u need to use substitution 5 time 
$$u=\dfrac{1}{x+2}$$
& then after that 
$$X=\sqrt{4u-1}$$
&
$$w=X+1$$
& 
....
but the final answer is $$\ln\left(\left|\sqrt{-\dfrac{x-2}{x+2}}-1\right|\right)-\ln\left(\sqrt{-\dfrac{x-2}{x+2}}+1\right)+2\arctan\left(\sqrt{-\dfrac{x-2}{x+2}}\right)+C$$
</s>