Question

Integrate ln(sec(x)) between x=0 and pi/4 hint: integral of sec(x) = ln(sec(x)+tan(x))

Please show me the steps rather than just the answer

Answer 1

pretty weird result from Mathematica
1/4 (-2 Catalan + \[Pi] Log[2])

the last part is familiar ...the first part is weird.

Answer 2

Weirder result from another one
-(7/96*I)*Pi^2+(1/8)*Pi*ln(2)-I*dilog(1-(1/2*I)*sqrt(2)+(1/2)*sqrt(2))+(1/8)*Pi*ln(2-sqrt(2))-(1/4*I)*Pi*arctan(sqrt(2)/(-2+sqrt(2)))+(1/8)*Pi*ln(2+sqrt(2))-(1/4*I)*Pi*arctan(sqrt(2)/(2+sqrt(2)))-I*dilog(1+(1/2*I)*sqrt(2)-(1/2)*sqrt(2))

Answer 3

http://en.wikipedia.org/wiki/Catalan's_constant

Answer 4

yeah i saw that one ... still it's hard to get that one.

Answer 5

-1/2 Integrate[Log[Cot[x]]+ 2 Log[Cos[2 x]], {x, 0, Pi/4}]
this would give the required value
just need to show that

-1/2 Log[Cot(x) + 2 Log(cos(2x))] = log(sec(x))

Answer 6

it seems they are different ...