If `int(1)/(1-sin ^(4)x)dx=(1)/(2) tanx+Atan^(-1){f(x)}+C`, then

Options

A. `A=(1)/(2sqrt(2))andf(x) =sqrt(2)tanx`
B. `A=sqrt(2)andf(x) =sqrt(2)tanx`
C. `A=-sqrt(2)andf(x) =sqrt(2)tanx`
D. none of these

Answer

A

Explanation

Let `I=int(1)/(1-sin^(4)x)dx=int(1)/((1-sin^(2)x)(1+sin^(2)x))dx`
`rArrI=int(sec^(2)x)/(1+sin^(2)x)dx=int(1+tan^(2)x)/(1+2tan^(2)x)d(tanx)`
`rArrI=(1)/(2)int(tan^(2)x+(1)/(2)+(1)/(2))/(tan^(2)x+(1)/(2))d(tanx)`
`rArrI=(1)/(2)int(1+(1)/(2)*(1)/(tan^(2)x+(1)/(2)))d(tanx)`
`rArrI=(1)/(2)[tan x +(sqrt(2))/(2)tan^(-1)(sqrt(2)tanx)]+C`
`rArrI=(1)/(2)tan x +(sqrt(2))/(4)tan^(-1)(sqrt(2)tanx)+C`
Hence , `A=(sqrt(2))/(4)=(1)/(2sqrt(2))andf(x)=sqrt(2)tanx`.