Prove that `(d^n)/(dx^n)(e^(2x)+e^(-2x))=2^n[e^(2x)+(-1)^n e^(-2x)]`

frac {d^ny}{dx^n} (x^2e^(2x))

Prove that: I_(n)=int_(0)^(oo)x^(2n+1)e^(-x^(2))dx=(n!)/(2),n in N

If o be the sum of odd terms and E that of even terms in the expansion of (x+a)^(n) prove that: O^(2)-E^(2)=(x^(2)-a^(2))^(n)( (i) 4OE=(x+a)^(2n)-(x-a)^(2n)( iii) 2(O^(2)+E^(2))=(x+a)^(2n)+(x-a)^(2n)

Differentiate (e^(2x)+e^(-2x))/(e^(2x)-e^(-2x)) with respect to x:

Differentiate (e^(2x)+e^(-2x))/(e^(2x)-e^(-2x)) with respect to 'x'

Differentiate (e^(2x)+e^(-2x))/(e^(2x)-e^(-2x)) with respect to 'x'

Show that Gamma(n)=2int_(0)^(infty)e^(-x^(2))x^(2n-1) dx.

The function f (x)={((x ^(2n)))/((x ^(2n) sgn x)^(2n+1))((e ^(1/x)-e ^(-1/x))/(e ^(1/x)+e ^(-(1)/(x))))x ne0 n in N is:

The function f (x)={((x ^(2n)))/((x ^(2n) sgn x)^(2n+1))((e ^(1/x)-e ^(-1/x))/(e ^(1/x)+e ^(-(1)/(x))))x ne0 n in N is: