`intsqrt((1-sqrt(x))/(1+sqrt(x)))dx` का मान ज्ञात कीजिए।

Explanation

माना `sqrt(x) = cos 2 theta`
`rArr (1)/(2sqrt(x)) dx = -2sin 2theta d theta `
`rArr" "dx = - 2sqrt(x).2sin 2 theta d theta = - 4cos2 theta sin 2theta d theta`
`thereforeint sqrt((1-sqrt(x))/(1+sqrt(x))) dx`
`= int sqrt((1-cos2theta)/(1+cos 2 theta)).(-4 cos 2 theta sin 2 theta )d theta`
`= int (sin theta)/(cos theta) (- 4 cos 2 theta . 2 sin theta cos theta) d theta`
`= - 8 int cos 2theta sin ^(2) theta d theta`
`8 int cos 2 theta.((1-cos 2theta))/(2) d theta`
`= - 4((sin 2theta)/(2) - int (1+cos 4theta)/(2) d theta)+c`
` = - 2(sin 2 theta - 2theta - (sin 4 theta)/(4))+c`
`=-2 sin 2 theta + 2theta +(1)/(2) . 2 sin 2 theta cos 2 theta + c`
` = - 2 sqrt(1-x)+cos^(-1)sqrt(x)+sqrt(1-x). sqrt(x+c)`