`12 + 9 sqrt((x-1)(3x+2))= 3x^2 -x`

Using properties of proportion, solve for x : (i) (sqrt(x + 5) + sqrt(x - 16))/ (sqrt(x + 5) - sqrt(x - 16)) = (7)/(3) (ii) (sqrt(x + 1) + sqrt(x - 1))/ (sqrt(x + 1) - sqrt(x - 1)) = (4x -1)/(2) . (iii) (3x + sqrt(9x^(2) -5))/(3x - sqrt(9x^(2) -5)) = 5 .

If int (x +(cos^-1 3x)^2)/sqrt(1 - 9x^2) dx = A sqrt(1 - 9x^2) + B (cos^-1 3x)^3 + c then A, B are respectively

If sqrt(3x^(2)-12x + 19) + sqrt(3x^(2)-12x-11)= 16 then find sqrt(3x^(2)-12x + 19) - sqrt(3x^(2) -12x-11) = ?

int frac {x+1}{x+2} dx =............ A) x+2 - log(x+2) + c B) sqrt (x^2 + 3x +2) -(1/2) log x + (3/2) + sqrt (x^2 + 3x +2) + c C) sqrt (x^2 + 3x +2) +(1/2) log x + (3/2) + sqrt (x^2 + 3x +2) + c D) sqrt (x^2 + 3x +2) -(1/2) log x + (3/2) - sqrt (x^2 + 3x +2) + c

If Delta (x) =|{:(4x-4,(x-2)^(2),x^(3)),(8x-4sqrt(2),(x-2sqrt(2)^(2)),(x+1)^(3)),(12x-4sqrt(3),(x-2sqrt(3))^(2),(x-1)^(3)):}| then

If Delta (x) =|{:(4x-4,(x-2)^(2),x^(3)),(8x-4sqrt(2),(x-2sqrt(2)^(2)),(x+1)^(3)),(12x-4sqrt(3),(x-2sqrt(3))^(2),(x-1)^(3)):}| then

int (x+ ( cos^(-1)3x )^(2))/(sqrt(1-9x ^(2)))dx = (1)/(k _(1)) ( sqrt(1-9x ^(2))+ (cos ^(-1) 3x )^(k_(2)))+c, then k _(1) ^(2)+k_(2)^(2)= (where C is an arbitrary constnat. )