∫ dy/cos 5y?

Answer 1

∫sec x dx = ln|sec x + tan x| + C

so

∫ dy/cos 5y = ∫ sec5y dy = ...

5y=u -> 5dy = du -> dy = 1/5du

... 1/5 ∫ secu du = 1/5 ln|sec u + tan u| + C =

= 1/5 ln|sec 5y + tan 5y| + C

Hope this helps!

Answer 2

I = ∫ (1/cos 5y).dy
Let u = 5y
du = 5.dy
dy = du / 5
I = (1/5) ∫ sec u du
This is designated as a standard integral given by:-
I = (1/5).log (sec u + tan u) + C
I = (1/5).log.(sec 5y + tan 5y) + C

Answer 3

∫ dy/cos5y

=∫ sec5y dy
=(log I sec5y+tan5y I)/5 +c