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.

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

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

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

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

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

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

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

If the normal at one end of lotus rectum of an ellipse passes through one end of minor axis then prove that, e^(4)+e^(2)-1=0