What is the net area between #f(x) = cscx -cosxsinx# and the x-axis over #x in [pi/6, (5pi)/8 ]#?

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Answer 1 · 2018-02-20T15:01:49

#int_(pi/6)^((5pi)/8)cscx-cosxsinxdxapprox1.4181#

We want to find #I=int_(pi/6)^((5pi)/8) cscx-cosxsinxdx#.

We do this by integrating term-by-term and then applying the limits.

#intcscxdx=-ln(cscx+cotx)#

This integral is well known

#int-cosxsinxdx#

Let #u=cosx# and #du=-sinxdx#

Then #int-cosxsinxdx=intudu=1/2u^2=1/2cos^2x#

So #I=[1/2cos^2x-ln(cscx+cotx)]_(pi/6)^((5pi)/8)approx1.4184#