I have the following definite integral:
$$\int_{E_{\theta}-\frac{\phi}{1+\lambda}}^{E_{\theta}}(E_{\theta}+\lambda(E_{\theta}-v))dv$$
Both $E_{\theta}$ and $\lambda$ are greater than $0$. Is there any way of determining whether the value of this definite integral is increasing in $E_{\theta}$, without actually differentiating it with respect to $E_{\theta}$? Presently, I claim that it is increasing in $E_{\theta}$, as the integrand is increasing in $E_{\theta}$, and the ‘length’ of the domain of integration does depend on $E_{\theta}$. Is this correct? Thank you.
Your proof works, but I feel the statement "the integrand is increasing in $E_θ$" needs more elaboration, since the values which $v$ takes on depends on $E_θ$
Here's how I prove it.
Make the substitution $u=v-E_\theta$, we see that the integral is equal to $\int_{\frac{-\phi}{1+\lambda}}^0 (E_\theta - \lambda u) du$.
Assuming $\phi>0$, this is increasing in $E_{\theta}$.