Mock Test 2 — Definite Integration

Evaluate the classical log-trigonometric definite integral $I = \int_0^{\pi/2} \ln(\sin x) \, dx$.

Solve the following infinite series product sequence limit: $L = \lim_{n \to \infty} \left[ \frac{n!}{n^n} \right]^{1/n}$.

Evaluate the fractional part definite integral $I = \int_0^{10} \{x\} \, dx$, where $\{\cdot\}$ denotes the fractional part function.

Find the value of the high-order transcendental limit expression: $\lim_{x \to 0} \frac{\int_0^{x^2} \cos t^2 \, dt}{x^2}$.

Evaluate the following challenging definite integral: $I = \int_0^{\pi} \frac{x}{1 + \sin x} \, dx$.

Evaluate the mixed algebraic floor integral form $\int_0^2 [x^2] \, dx$, where $[\cdot]$ is the greatest integer function.

Find the value of the improper integral transformation form $I = \int_0^\infty \frac{\ln x}{1+x^2} \, dx$.

Evaluate the product trigonometric integral $I = \int_0^{\pi/2} \sin^4 x \cos^2 x \, dx$ using Wallis extensions.

Let $f(x)$ be a continuous function satisfying $f(x) + f(a-x) = k$. The value of $\int_0^a f(x) \, dx$ is:

Evaluate the parameter differentiation form integral (Feynman's Technique) $I(\alpha) = \int_0^1 \frac{x^\alpha - 1}{\ln x} \, dx$ for $\alpha > 0$.

Find the fundamental period $T$ of the trigonometric baseline expression $f(x) = \sin x + \cos x$.

Find the value of the integer parameter $k$ if the definite integral satisfies $\int_{-\pi}^{\pi} |x| \sin x \, dx = k$.

If $\int_0^\infty t^n e^{-t} \, dt = 120$, find the positive integer value of the exponent parameter $n$.

Assertion (A): The periodic definite integral equation satisfies $\int_0^{10\pi} |\sin x| \, dx = 10 \int_0^\pi \sin x \, dx$.
Reason (R): The absolute sine function $|\sin x|$ is periodic with a fundamental period of exactly $\pi$, allowing the upper boundary multiplier to be factored out.

Assertion (A): The improper integral $\int_1^\infty \frac{1}{x^2} \, dx$ is convergent and evaluates to exactly 1.
Reason (R): According to the standard p-test criteria, rational infinite integrals converge if and only if the exponent satisfies $p > 1$.