Question

Evaluate: `int1/(sin^4x+cos^4x)dx`
A. `(1)/(sqrt(2))tan^(-1)((tan2x)/(sqrt(2)))+C`
B. `(1)/(sqrt(2))tan^(-1)((1+cos2x)/(sqrt(2)))+C`
C. `(1)/(sqrt(2))tan^(-1)((tanx+cotx)/(sqrt(2)))+C`
D. `sqrt(2)tan^(-1)((sqrt(tan)x+sqrt(cot)x)/(sqrt(2)))+C`

Answer 1

Correct Answer - a
We have , `sinx+cos^(4)x=((1-cos2x)/(2))^(2)+((1+cos2x)/(2))^(2)=(1)/(2)(1+cos^(2)2x)`
`:. I=int(1)/(sin^(4)x+cos^(4)x)dx`
`rArrI=2int(1)/(1+cos^(2)2x)dx=2int(sec^(2)2x)/(sec^(2)2x+1)dx`
`rArrI=int(2sec^(2)2x)/(2+tan^(2)2x)dx=int(1)/(2+tan^(2)2x)d(tan2x)`
`rArrI=(1)/(sqrt(2))tan^(-1)((tan2x)/(sqrt(2)))+C`