Question

Q.1 ARE THE FUNCTIONS EVEN OR ODD

1- INTEGRAL IS FROM MINUS INFINITY TO PLUS INFINITY
∫e^(-2x)(1-e^(-x))^(2) dx

2-INTEGRALIS FROM MINUS INFINITY TO PLUS INFINITY
∫x(e^(-2x))(1-e^(-x))^(2) dx

Answer 1

1.

$f(x)=_{-\infty}^{\infty}\int e^{-2x}(1-e^{-x})^2dx$

Changing x with -x we get;

$f(-x)=_{-\infty}^{\infty}\int e^{2x}(1-e^{x})^2dx$

Now $f(x)+f(-x) \neq 0$

And also $f(x)-f(-x) \neq 0$

Thus, f(x) is neither even nor odd.

2. Let $g(x)=_{-\infty}^{\infty}\int x e^{-2x}(1-e^{-x})^2dx$

Changing x with -x, we get;

$g(-x)=_{-\infty}^{\infty}\int- x e^{2x}(1-e^{x})^2dx$

Now, $g(x)+g(-x) \neq 0$

And also $g(x)-g(-x) \neq 0$

Thus, g(x) is also neither even nor odd.