cosx/(sinx + √sinx) integrate
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Integ.sin x/(sin x + cos x) dx
On dividing both numerator and denominator with cos x
Integ.tan x/(1+tan x) dx
Let tan x =t. Sec^2 x dx= dt ,.
So dx =dt/(1+tna^2 x) . ie dx= dt/(1+t^2).
The sum becomes inreg. t dt(1+t)(1+f^2)
= (1/2)integ. [{(1+t)/(1+t^2)} +1/(1+t)] dt
=integ. (1/2).1/1+t^2) dt + integ.(1/4).2t/1+t^2) dt +integ. (1/2).1/1+t) dt
=1/2. tan^-1 t + 1/4.log(1+t^2) + 1/2 log(1+t) + c
=(1/2).tan^-1 tan x (1/4) log (1+tan^2 x) + (1/2)log(1+tan x) + c
On dividing both numerator and denominator with cos x
Integ.tan x/(1+tan x) dx
Let tan x =t. Sec^2 x dx= dt ,.
So dx =dt/(1+tna^2 x) . ie dx= dt/(1+t^2).
The sum becomes inreg. t dt(1+t)(1+f^2)
= (1/2)integ. [{(1+t)/(1+t^2)} +1/(1+t)] dt
=integ. (1/2).1/1+t^2) dt + integ.(1/4).2t/1+t^2) dt +integ. (1/2).1/1+t) dt
=1/2. tan^-1 t + 1/4.log(1+t^2) + 1/2 log(1+t) + c
=(1/2).tan^-1 tan x (1/4) log (1+tan^2 x) + (1/2)log(1+tan x) + c
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