evaluate the integral I= ∫(0,∏/4) ((5-3e^(-tanx))/(cos^(2)x))dx

Answer 1

write it as

5/cos^2 x - 3e^(-tanx)/cos^2x
= 5sec^2x - 3e^(-tanx)/cos^2 x
= 5sec^2x - 3e^(-tanx) (sec^2 x)

now for recognition facts
recall that d(tanx)/dx = sec^2x

so the integral of 5sec^2x is 5tanx
for the integral of the exponential, remember that when we differentiate an e^(anything) function, the original e^anything comes back, times the derivative of "anything"
we have exactly that pattern

so the integral of -3e^(-tanx) (sec^2 x) is
+ 3e^(-tanx)

so ∫((5-3e^(-tanx))/(cos^(2)x))dx
= 5tanx + 3e^(-tanx) + a constant

I will leave the substitution up to you