∫ calculus

1. The problem is to find the derivatives of logarithmic and exponential functions. 2. The derivative of the natural logarithm function $\ln(x)$ is given by the formula:

1. **Stating the problem:** Calculate the definite integral $$\int_0^2 x^3 \, dx$$ using the Fundamental Theorem of Calculus. 2. **Formula and rules:** The Fundamental Theorem of C

1. Задача: минимизировать функцию $f(x) = 2x^2 - e^x$ на интервале $[0,1]$. 2. Для нахождения минимума функции используем производную: $f'(x) = 4x - e^x$.

1. Problem: Find the limit $$\lim_{x \to 0} \frac{\cos x}{x+1}$$. 2. Formula and rules: The limit of a quotient is the quotient of the limits if both limits exist. Also, $$\lim_{x

1. Задача: минимизировать функцию $$f(x) = 2x^2 - e^x$$ на интервале $$[0,1]$$. 2. Для минимизации функции необходимо найти её критические точки, где производная равна нулю, и пров

1. ปัญหาคือการหาค่าของฟังก์ชัน $f(x,y) = xy + 2x^{2} - 3y + 27$ โดยที่ $x$ และ $y$ เป็นฟังก์ชันของตัวแปร $t$ คือ $x(t) = t^{2} - 2t - 3$ และ $y(t) = 4t - 3$ และต้องการหาค่าที่ $t =

1. **Problem Statement:** Find the area of the region enclosed by the curves $y = (x - 2)^2$ and $y = 4 + 4x - x^2$. 2. **Find the points of intersection:** Set the two functions e

1. **State the problem:** We want to prove that for $0 < a < b < 1$, the inequality $$\frac{b - a}{\sqrt{1 - a^2}} < \sin^{-1} b - \sin^{-1} a < \frac{b - a}{\sqrt{1 - b^2}}$$

1. **State the problem:** Evaluate the limit $$\lim_{x \to 0} (1 + \tan x)^{\cot x}$$. 2. **Recall the formula and rules:** This is a limit of the form $$\lim_{x \to 0} (1 + f(x))^

1. **State the problem:** We need to find the integral of the function $y = 3x^2 + 2x - 5$ with respect to $x$. 2. **Recall the formula:** The integral of a polynomial term $ax^n$

1. Statement of the problem. We are given the function

1. Problem: Differentiate the function $k(x)=x[\sin(\ln x)-\cos(\ln x)]$. 2. Formula: Use the product rule and the chain rule.

1. The problem is to find the area under the curve of the function $$y = -0.015x^2 + 60$$ over a certain interval. 2. To find the area under a curve, we use the definite integral o

1. **Problem:** Find the derivative of the function $f(x) = (2x^2 - 3x)(-4x^{-2} + 5)$. 2. **Formula and rules:** Use the product rule for derivatives:

1. **State the problem:** Evaluate the line integral $$\int_C y^2 \, dx + x^2 y \, dy$$ where $C$ is the rectangle with vertices $(0,0)$, $(5,0)$, $(5,4)$, and $(0,4)$ oriented cou

1. Diberikan fungsi $f(x) = e^{-x^2}$. Kita diminta mencari turunan ke-10 dari fungsi ini. 2. Fungsi ini adalah fungsi Gaussian, dan turunannya dapat dihitung menggunakan aturan ra

1. **بيان المسألة:** نريد مقارنة الدالتين $f(x)$ و $g(x)$ في المجالات المعطاة.

1. **Problem statement:** Given the function $$g(x) = x - 1 + \ln(x)$$ defined on $$]0, +\infty[,$$ prove that its derivative is $$g'(x) = 1 + \frac{1}{x}$$ for all $$x \in ]0, +\i

1. Diberikan limit $$\lim_{x \to \infty} \left(\sqrt{4x^2 - 2x + 6} - \sqrt{4x^2 + 2x - 1}\right)$$. 2. Untuk menyelesaikan limit ini, kita gunakan trik mengalikan dengan konjugat

1. The problem is to understand how the integrand was derived step by step. 2. An integrand is the function inside an integral sign that we want to integrate.

1. The problem asks how the integrand was obtained for parts 1, 2, and 3. 2. The integrand is the function inside the integral sign that we integrate with respect to a variable.