[" 33."log(log i)=],[[" 1) "log(pi)/(2)," 2) "(pi)/(2)log i],[" 3) "log(pi)/(2)+(i pi)/(2)," 4) "log(pi)/(2)-i(pi)/(2)]]

[int_(0)^(0)cot^(-1)(1-x+x^(2))dx=],[[" (A) "(pi)/(2)-ln2," (B) "(pi)/(2)+ln2],[" (C) "(pi)/(2)-ln sqrt(2)," (D) "(pi)/(2)+ln sqrt(2)]]

[int_(-pi/4)^( pi/4)log(cos x+sin x)dx=(1)/(2)],[[" 1) "pi log2," 2) "-pi log2," 3) "-(pi)/(4)log2," 4) "-(pi)/(2)log2]]

int_(0)^((pi)/(2))log(sinx)dx=int_(0)^((pi)/(2))log(cosx)dx=(pi)/(2)log.(1)/(2)

int_(0)^((pi)/(2))log(sinx)dx=int_(0)^((pi)/(2))log(cosx)dx=(pi)/(2)log.(1)/(2)

int_(0)^((pi)/(2))log sin xdx=int_(0)^((pi)/(2))log cos xdx=(1)/(2)(pi)log((1)/(2))

Show that int_(0)^((pi)/(2))log(sin2x)dx=-(pi)/(2)(log2)

f(x)={((5^(cosx)-1)/((pi)/(2)-x)",", x ne (pi)/(2)), (log 5"," , x =(pi)/(2)):} at x =(pi)/(2) is

f(x)={((5^(cosx)-1)/((pi)/(2)-x)",", x ne (pi)/(2)), (log 5"," , x =(pi)/(2)):} at x =(pi)/(2) is

Show that int_(0)^((pi)/(2))logsinxdx=(pi)/(2)log((1)/(2))=(-pi)/(2)log2

int_(0)^((pi)/(4))log(sin2 theta)d theta=-((pi)/(4))log((1)/(2))