`intsqrt(1+secx)dx`

Options

A. ` sin ^( - 1 ) (sqrt 2 sin x ) +C `
B. `2 sin ^( - 1) (sqrt 2sin x// 2) + C `
C. ` 2 sin ^( -1 ) (sqrt 2 sinx ) + C `
D. ` 2 sin ^( - 1 ) (sqrt2 x // 2 ) + C `

Answer

B

Explanation

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