How do you evaluate the definite integral #int tant# from #[0,pi/4]#?

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Answer 1 · 2016-10-16T18:29:31

#= 1/2ln 2 #

#int_0^(pi/4) tant \ dt#

#= int_0^(pi/4) (sin t)/( cos t) \ dt#

spotting the pattern
#= int_0^(pi/4) (-(d/dt)(cos t))/( cos t ) \ dt#

and the other pattern
#= - int_0^(pi/4) (d/dt) ln ( cos t )\ dt#

reversing the integration interval because of the -ve sign

#= [ ln ( cos t ) ]_(pi/4)^0#

#= ln ( 1) - ln (1/sqrt 2) #

#= 1/2ln 2 #