integration by substitution

Question

Transform this integral using the substitution u = cos x.

∫ upper limit 𝜋/3, lower limit 𝜋/4 tan x dx

Your final answer should be a new integral written in terms of u:

_____________

Answer 1

tan = sinx/cosx

u = cosx

du =-sinxdx

sinxdx/cosx = -du/u

integral of -du/u = -ln|u| + c

where c = an unknown constant

convert the limits from x to u

u=cosx

limit = u = cos(pi/4) =cos45=(sqr2)/2 = about .707

limit = u =cos(pi/3) = cos60 = 1/2 =.5

evaluated from x=pi/4 to pi/3

is the same as from u=about .707 to u = .5

-ln(.707) - (-ln.5)

=-ln(.707) -(ln(1) -ln(2))

= ln(.707) - (0 - ln2)

= ln(.707) +ln2

= ln(2(.707))

= ln(1.414)

=ln(sqr2)

=about .3466

integral of tanx = -ln|cosx| + c

= ln|secx| + c

evaluated from pi/4 to pi/3

= ln(secpi/3) -ln(secpi/4)

=ln[(1/cospi/3) - ln(1/cospi/4)

=ln(1/1/2) - ln(1/sqr2/2)

= ln2 - ln(2/sqr2)

=ln2 -ln(sqr2)

= ln(2/sqr2)

= ln(sqr2)

=about 0.3466