How do you evaluate the definite integral $\int tan t$ $int tant$ from $[0, \frac{π}{4}]$ $[0,pi/4]$ ?

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Answer 1 · 2016-10-16T18:29:31

$= \frac{1}{2} ln 2$ $= 1/2ln 2$

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

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

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

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

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

$= 1/2ln 2$

How do you evaluate the definite integral int tant∫tantint tant from [0,pi/4][0,π4][0,pi/4]?