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

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 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)

If e_(1)ande_(2) be the eccentricities of the ellipses (x^(2))/(a^(2))+(y^(2))/(b^(2))=1and(x^(2))/(a^(2))+(4y^(2))/(b^(2))=1 respectively then prove that 3=4e_(2)^(2)-e_(1)^(2) .

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

A B C is a triangle. D is a point on A B such that A D=1/4A B and E is a point on A C such that A E=1/4A Cdot Prove that D E=1/4B Cdot

A B C is a triangle. D is a point on A B such that AD=1/4A B\ a n d\ E is a point on A C such that A E=1/4A C . Prove that D E=1/4B C