prove it `2e^(-1/4) < int_0^2e^(x^2-x)dx < 2e^2`

If the normals at an end of a latus rectum of an ellipse passes through the other end of the minor axis, then prove that e^(4) + e^(2) =1.

Prove that 2tan^(-1)((1)/(2))=tan^(-1)((4)/(3))

Prove that tan^(-1) (1/4) + tan^(-1) (2/9) = 1/2 sin^(-1) (4/5)

Prove that (tan^(-1)(1)/(e))^(2)+(2e)/((e^(2)+1))<(tan^(-1)e)^(2)+(2)/(sqrt(e^(2)+1))

Prove that (1/x)^(x) has maximum value is (e)^(1/e) .

Prove that e^(x) ge 1 +x and hence e^(x) +sqrt(1+e^(2x))ge(1+x)+sqrt(2+2x+x^(2)) forall x in R

Prove that tan^-1(1/4)+ tan^-1(2/9) = 1/2sin^-1(4/5)

Prove that 1^2/(1!)+2^2/(2!)+3^2/(3!)+4^2/(4!)+.....=2e

Prove that 1^3/(1!)+2^3/(2!)+3^3/(3!)+4^3/(4!)+.....=5e

If e and e' the eccentricities of a hyperbola and its conjugate,prove that (1)/(e^(2))+(1)/(e^(2))=1