`frac { 1 } { 1 + sqrt { 2 } } `

Write the cot^-1 (frac {1}{ sqrt(x^2 -1) }), |x| > 1 . in the simplest form.

Evaluate: \int_{0}^1 frac{1}{sqrt (3+ 2x - x^2)} dx

If y= tan^(-1) (frac{x}{1+ sqrt (1 - x^2)}) + sin[ 2tan^(-1) (sqrt ((1-x)/(1+x)))] , then dy/dx= ............

If y= tan^(-1) (sqrt (frac {a-x}{a+x})) , where -a < x < a then dy/dx = ........ A) frac{x}{sqrt (x^2 + a^2)} B) frac{a}{sqrt (a^2 + x^2)} C) frac{1}{2 sqrt (a^2 - x^2)} D) -frac{1}{2 sqrt (a^2 - x^2)}

If the shortest distance between the lines frac{x - lambda} {-2} = frac{y - 2} {1} = frac{z - 1} {1} and frac{x - sqrt 3}{1} = frac{y-1}{-2} = frac{z-2}{1} is 1, then the sum of all possible values of lambda is :

If y= sin^(-1) (frac{5x+ 12 sqrt (1-x^2)}{13}) , then (dy/dx) = .......... A) 0 B) frac {-1}{sqrt (1-x^2)} C) frac {1}{sqrt (1-x^2)} D) sqrt (1-x^2)

int frac{x^2 + 3x +5}{(x+2)(x^2 + 2x +3)} dx=............ (A) log (x+2) + tan^(-1) (frac{x+1}{sqrt 2}) + c (B) log ( x+2) + logx^2 + 2x + 3 + c (C) log (x+2) -frac{1}{ sqrt 2} tan^(-1) (frac{x+1}{sqrt 2}) + c (D) log(x+2) +frac{1}{ sqrt 2 } tan^(-1) (frac{x+1}{sqrt 2}) + c

If y= sin(2 sin^(-1) x) , then dy/dx= ............ A) frac{2-4x^2}{sqrt (1-x^2)} B) frac{2+4x^2}{sqrt (1-x^2)} C) frac{4x^2 - 1}{sqrt (1-x^2)} D) frac{1-2x^2}{sqrt (1-x^2)}

Differentiate tan^(-1) frac{ sqrt (1+ x^2 ) - 1}{x} w. r. t. cos^(-1) ( sqrt ( frac{1+ sqrt (1+ x^2 ) }{2 sqrt (1+ x^2 ) } ) )

Show that (frac{1}{sqrt 2}+frac{1}{sqrt 2}i )^10 + (frac{1}{sqrt 2}-frac{1}{sqrt 2}i )^10 = 0