Real Analysis

Find the derivative with respect to $x$ of the integral from 0 to $x$ of the function $\sqrt{t^2 + 4} \, dt$.

Evaluate the integral of $\sqrt{t^3 + 5}$ from $x$ to 4 and find its derivative.

Evaluate the integral from 5 to $x^2$ of $\sqrt{t^3 - 4}$ and differentiate it using the chain rule.

Integrate the function from $x^2$ to $x^3$ where the function is $a0 t{t^4 - 2}$ and find its derivative applying the fundamental theorem.

Compute the derivative of the integral from 3 to x of $\sin(T^2)$ with respect to $x$.

Find the derivative of the integral from 0 to x of $\frac{T}{1 + T^3}$ with respect to $x$.

Determine the infimum and supremum of the natural numbers.

Determine the infimum and supremum of the real numbers.

Determine the supremum and infimum of the set $\left\{ \frac{1}{n} : n \in \mathbb{N} \right\}$.

Determine the supremum and infimum of the set of all rational numbers whose square is less than two.

Prove that if a subset of $\R^P$ is compact, then it is closed and bounded.

Prove that a closed interval [c, d] of real numbers is a compact set.

Show that if a set $A$ is bounded and closed, then it is compact, according to the Heine-Borel theorem.

Prove that if a sequence is convergent, meaning it has a limit, then the lim sup of the sequence is equal to its limit.

Is the set of even integers countable?

Given an output value in a one-to-one correspondence between natural numbers and even integers, determine the input value.

Prove that the cardinality of the natural numbers, denoted as $\aleph_0$, is equal to the cardinality of the set of all positive odd integers.

Let C be the set of all integers n such that n = 6r - 5 for some integer r. Let D be the set of all integers m such that m = 3s + 1 for some integer s. Prove or disprove: (a) C is a subset of D; (b) D is a subset of C.

Prove that set A, which consists of all integers that can be written as $4p$, is a subset of set B, which consists of all integers that can be written as $2q$, where $p$ and $q$ are integers.

Interchange the limit of the function and solve the limit  lim_{x \to 0} \frac{\arcsin(x)}{1 + \sqrt{x}}.