`int sin(2x)/(sin^4x+cos^4x) dx`

Options

A. ` tan ^( - 1 ) (cot ^ 2 x ) + C `
B. ` - tan ^( - 1 ) ( cos 2x ) + C `
C. ` tan ^( - 1 ) ( sin 2x ) + C `
D. ` tan ^( -1 ) ( tan ^ 2 x) +C `

Answer

B

Explanation

Let I ` = int ( sin 2x ) /( sin ^ 4 x + cos ^ 4 x ) dx `
` I = int ( sin 2x )/( ( sin ^ 2 x + cos ^ 2 x ) ^ 2 - 2 sin ^ 2 x * cos ^ 2 x ) dx `
I ` = int ( sin 2x )/ ( 1 - (1 )/ (2 ) ( sin 2x )^ 2 ) dx `
` = 2 int ( sin 2 x ) /( 1 + ( 1 - sin ^ 2 2x ) ) dx `
(let ` t = cos 2x rArr dt = - 2 sin 2x dx `
` = int ( -dt )/( 1 + t ^ 2 ) `
` =- tan ^( - 1) t + C `
` =-tan ^( - 1 ) ( cos 2x) + C `