cosec ⁻¹ ( a+x x ) , d x =

Step-by-step solution

Let I = cosec ⁻¹ ( a+x x ) , d x . Substitute x = a ^2 , which gives = ⁻¹ ( x a ) . Differentiating both sides, we get d x = 2a ^2 , d . The term inside the inverse trigonometric function simplifies as: a+x x = a+a ^2 a ^2 = 1+ ^2 ^2 = ^2 ^2 = cosec Therefore, the integrand becomes cosec ⁻¹( cosec ) = . The integral transforms to: I = (2a ^2 ) , d Applying integration by parts, taking as the first function and 2a ^2 as the second function: I = 2a ^2 , d - ( d d ( ) 2a ^2 , d ) d Since 2a ^2 , d = a ^2 , we get: I =