Integrate 1÷(1 + cosec x ) ?

2 Answers
Apr 4, 2018

#int1/(1+cscx)dx=secx-tanx+x+c#

Explanation:

#int1/(1+cscx)dx#

= #int(cscx-1)/((cscx-1)(cscx+1))dx#

= #int(cscx-1)/(csc^2x-1)dx#

= #int(cscx-1)/cot^2xdx#

= #int(sin^2x/(sinxcos^2x)-tan^2x)dx#

= #int(sinx/cos^2x-sec^2x+1)dx#

= #int(secxtanx-sec^2x+1)dx#

= #secx-tanx+x+c#

Apr 4, 2018

#int1/(1 + cosec x )#

#=int1/(1 + 1/sin x )#

#=intsin x /(sin x + 1)#

#=int(sin x(sin x - 1)) /((sin x + 1)(sin x - 1))#

#=int(sinx-sin^2x)/(cos^2x)dx#

#=inttanxsecxdx-inttan^2xdx#

#=inttanxsecxdx-int (sec^2x-1)dx#

#=secx-(tanx-x)+c#

#=x+secx-tanx+c#