Prove that: `∫dx/(x^2 + a^2) = 1/a tan^(-1) (x/a) + C`

Assertion (A) : int(dx)/(x^(2) + 2x + 3) = (1)/(sqrt(2)) tan^(-1)((x+1)/(sqrt(2))) + c Reason (R) : int(dx)/(x^(2) + a^(2)) = (1)/(a) tan^(-1)((x)/(a)) + c

Prove that tan (2 tan^(-1) x ) = 2 tan (tan^(-1) x + tan^(-1) x^(3)) .

Prove that : 2 tan^(-1) (cosec tan^(-1) x - tan cot^(-1) x) = tan^(-1) x

Prove that : tan^(-1) x + cot^(-1) (1+x) = tan^(-1) (1+x+x^2)

Prove that : tan^(-1)x +tan^(-1). (2x)/(1-x^(2)) = tan^(-1) . (3x-x^(3))/(1-3x^(2)) , |x| lt 1/(sqrt(3))

int(1)/(x^(2)+a^(2))dx=(1)/(a)tan^(-1)((x)/(a))+c

Prove that tan^(-1) x + cot^(-1) (x+1) = tan ^(-1) (x^(2) + x+1) .

If int (dx)/((x^(2)+a^(2))^(2))=(1)/(ka^(2)){(x)/(x^(2)+a^(2))+(1)/(a) tan^(-1). (x)/(a)}+C . Then the value of k, is

Prove that 2 tan^(-1) (cosec tan^(-1) x - tan cot^(-1) x) = tan^(-1) x (x != 0)

Prove that (d)/(dx){2x tan^(-1)x-log (1+x^(2))}=2 tan^(-1)x.