Question

Integrate (a) 1/cos x (b) 1/sinx (c) 1 / 1 + cos x

Answer 1

a) ln(1/cosx + tanx) + c

b) ln | 1/tan(x/2) | +c

c)( 1/sinx ) - cotx +c

Answer 2

(1) integration of 1/cosx

dx/cosx = secx.dx
this is fit for basic integration formula,

= ln | secx + tanx | + C

=ln | 1/cosx + sinx/cosx | + C

=ln | 1 + sinx | - ln | cosx | + C


(2) Integration of 1/sinx

dx/sinx = cosecx.dx

=ln|cosecx - cotx | + C

= ln | 1 - cosx | - ln | sinx | + C

= ln | 2sin²x/2 | -ln | 2sinx/2.cosx/2 | + C

=ln | tanx/2 | + C


(3) 1/( 1 + cosx )

= dx/( 1 + cosx )

= dx/(2cos²x/2 )

=1/2 { sec²x/2 } dx

= tanx/2 + C