2. Evaluate f (--) dχ1+tax

1.	2. Evaluate f (--) dχ1+tax
Answer» I = ∫ [ 1 / ( 1 + tan x ) ] dx = ∫ { 1 / [ 1 + ( sin x / cos x ) ] } dx = ∫ [ cos x / ( sin x + cos x ) ] dx = (1/2) • ∫ [ ( 2 cos x ) / ( sin x + cos x ) ] dx ... Note This Step = (1/2) • ∫ { [ ( sin x + cos x ) + ( cos x - sin x ) ] / ( sin x + cos x ) } ... Note This Too = (1/2) • [ ∫ 1 dx + ∫ ( 1/u ) du ], ... u = sin x + cos x = (1/2) [ x + ln \|u\| ] + C = (1/2) [ x + ln \| sin x + cos x \| ]+C

Answer»

I = ∫ [ 1 / ( 1 + tan x ) ] dx

= ∫ { 1 / [ 1 + ( sin x / cos x ) ] } dx

= ∫ [ cos x / ( sin x + cos x ) ] dx

= (1/2) • ∫ [ ( 2 cos x ) / ( sin x + cos x ) ] dx ... Note This Step

= (1/2) • ∫ { [ ( sin x + cos x ) + ( cos x - sin x ) ] / ( sin x + cos x ) } ... Note This Too

= (1/2) • [ ∫ 1 dx + ∫ ( 1/u ) du ], ... u = sin x + cos x

= (1/2) [ x + ln |u| ] + C

= (1/2) [ x + ln | sin x + cos x | ]+C

Discussion