Regarding your first question: The theorem justifying the interchange of the limit and the integral is usually called something like "Leibniz' rule for differentiating under the integral sign". Assuming the integral limits are fixed (or at least: not functions of the differentiation variable $a$), it is sufficient that the integrand itself is continuous as a function of $x$ and $a$ and continuously differentiable w.r.t. $a$.Yashasvi said:What is the justification for differentiating some integrals with respect to constants in order to obtain result, i.e. ∂/∂a[∫e^(− ax^2).dx] =∫-x^2.e^(-ax^2) dx?I mean what if we say "a" was 3 then differentiating wrt 3 would have no significance?How can we treat it like a multivariable function :/
"Differentiating wrt constant" means taking the derivative of a function with respect to a constant term. This means that the constant term is treated as a fixed value and does not change when the derivative is taken.
Differentiating wrt constant helps us to simplify the integral and make it easier to solve. It allows us to treat the constant as a separate term and evaluate the integral without worrying about its value changing.
To differentiate wrt constant, simply treat the constant as a separate term and use the rules of differentiation to find the derivative of the function. The derivative of a constant is always 0, so it can be omitted from the final answer.
Sure, let's say we have the integral ∫(x + 3)dx. We can differentiate wrt constant by treating 3 as a constant term and finding the derivative of x, which is 1. This simplifies the integral to ∫(x + 3)dx = x^2/2 + 3x + C, where C is the constant of integration.
If you don't differentiate wrt constant, you may end up with a more complicated integral that is harder to solve. This can make the integration process longer and more difficult, and may result in a more complex final answer.