∫ calculus

1. The problem is to find the first and second derivatives of the function $f(x) = x^3 - 5x^2 + x - 1$. 2. Recall the power rule for derivatives: if $f(x) = x^n$, then $f'(x) = nx^

1. **Problem statement:** Show that for all $x > 0$, the inequality $$\frac{x}{1+x^2} < \tan^{-1} x < x$$ holds. 2. **Recall the functions and their properties:**

1. Problem: Compute $\frac{d}{dx} (3 \sin x - 4 \cos x)$.\n\n2. Formula: The derivative of $\sin x$ is $\cos x$, and the derivative of $\cos x$ is $-\sin x$.\n\n3. Apply the deriva

1. **Problem statement:** Find the Laplace transform of the function $f(t) = t \sqrt{\sin 2t}$. 2. **Recall the Laplace transform definition:**

1. Задача: вычислить двойной интеграл $$\int_0^\infty \int_0^{\frac{\pi}{2}} \frac{x \sin(\theta) \ln(1 + x^2 \cos^2(\theta))}{(1 + x^2 \sin^2(\theta))^{3/2}} \, d\theta \, dx$$. 2

1. **State the problem:** We need to solve the integral $$\int \frac{x^3 + 1}{x^5} \, dx$$. 2. **Rewrite the integrand:** Split the fraction into separate terms:

1. Énonçons le problème : Calculer l'intégrale $$\int \frac{x}{\sqrt{4 - x^2}} \, dx$$. 2. Utilisons la substitution recommandée : posons $$t = \sqrt{4 - x^2}$$.

1. **State the problem:** We need to evaluate the integral $$\int \frac{e^{2x}}{e^x + 3} \, dx.$$\n\n2. **Rewrite the integral:** Notice that $e^{2x} = (e^x)^2$. So the integral be

1. The problem asks: What is a limit in mathematics? 2. A limit describes the value that a function approaches as the input (or variable) approaches some point.

1. The problem is to understand the definition of the derivative and its applications such as instantaneous velocity and tangent lines. 2. The derivative of a function $f$ at a poi

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $y = 3x^3 + 4x^2 - 2x^{-1}$.\n\n2. **Recall the differentiation rules:**\n- Power rule: $\frac{d}{dx}

1. The problem is to understand why $$e^{1 - \infty} = e^{-\infty} = 0$$ when evaluating the limit of a function as $x \to 0^+$. 2. Recall the properties of exponents and limits: w

1. The problem is to understand why the limit $$\lim_{x \to 0^+} e^{1-\infty}$$ equals 0 and not negative infinity. 2. Recall the properties of limits and exponentials: when the ex

1. **Problem:** Express the integral $$\int_0^1 (2x - 1)^3 \, dx$$ in terms of the variable $$u = 2x - 1$$ without evaluating it. 2. **Formula and substitution rule:** When substit

1. Stating the problem: We need to find the limit of the function $$F(x) = x + 1 - \sqrt{x^2 - x - 2}$$ as $$x$$ approaches infinity. 2. Important formula and rules: When dealing w

1. Let's start by stating a common problem: Find the derivative of the function $f(x) = x^3 + 5x^2 - 4x + 7$. 2. The formula for the derivative of a power function $x^n$ is given b

1. The problem asks to find the sign of the function $f$ given its derivative $f'(x) = \frac{x^2 - 4x + 3}{(x-2)^2}$. 2. To determine where $f$ is increasing or decreasing, we anal

1. **State the problem:** We need to find the limit \( \lim_{x \to -2} \frac{8 + x^3}{4 - x^2} \). 2. **Recall the limit evaluation method:** If direct substitution leads to a defi

1. The problem asks to calculate three types of Riemann sums for the function $g(x)$ on given intervals: A. Right Riemann sum $R_4$ on $[0,2]$

1. **Problem Statement:** Calculate the definite integrals of the function $f(x)$ whose graph consists of semicircles below and above the x-axis on given intervals. 2. **Understand

1. The problem is to find the limit: $$\lim_{x \to -2} \frac{4 - x^2}{2 + x}$$. 2. We start by checking if direct substitution is possible by plugging in $x = -2$: