int_(0)^(oo)f((a)/(x)+(x)/(a))(log x)/(x)dx=f(a)int_(0)^(oo)f((a)/(x)+(x)/(a))(dx)/(x)

If I_(1)=int_(0)^(oo)f((2)/(x)+(x)/(2))(ln x)/(x)dx,I_(2)=int_(0)^(oo)f((2)/(x)+(x)/(2))(dx)/(x) and I_(3)=int_(-oo)^( oo)xf(e^(x)+e^(x))dx then

int_(0)^(oo)f((a)/(x)+(x)/(a))*(ln x)/(x)dx=ln a*int((a)/(x)+(x)/(a))*(dx)/(x)

int_(0)^(oo)(a^(-x)-b^(-x))dx

int_(0)^(a)(f(x))/(f(x)+f(a-x))dx=

int_(0)^(oo)(a^(-x)-b^(-x))dx=

int_(0)^(oo) (ln x)/(x^(2)+a^(2))dx=

int_(0)^(a) (f(x)+f(-x))dx=

int_(0)^(a)[f(x)+f(a-x)]dx=

int_(0)^(a) (f(a+x) + f(a-x) ) dx =

" 8."int_(0)^(oo)(log x)/(1+x^(2))dx