∫ calculus

1. **State the problem:** Calculate the integral $$\int \frac{1 + x + x^2}{x^4} \, dx$$. 2. **Rewrite the integrand:** Divide each term in the numerator by $x^4$:

1. Problem: Find the area bounded by the curves $y^2 = 9x$ and $y = 3x$. 2. Formula and rules: The area between two curves $y=f(x)$ and $y=g(x)$ from $x=a$ to $x=b$ is given by $$\

1. مسئله را بیان می‌کنیم: باید تابع $f(x)$ را پیدا کنیم که در معادله $$\int \frac{(\sqrt{x} - 1)(x + \sqrt{x})}{x^2} \, dx = \frac{1}{\sqrt{x}} f(x) + C$$

1. مسئله: یافتن تغییر متغیر مناسب برای انتگرال $$\int \frac{dx}{\sqrt{x} + \sqrt[3]{x}}$$ است. 2. برای ساده کردن عبارت مخرج، باید تغییر متغیری انتخاب کنیم که توان‌های کسری $\frac{1

1. The problem is to evaluate the limit $$\lim_{x \to 1} f(x)$$ without necessarily drawing the graph. 2. When evaluating limits, especially as $x$ approaches a specific value, you

1. The problem asks to find the limit \(\lim_{x \to 1} 4x - 3\) using a graph. 2. The function is \(y = 4x - 3\), which is a linear function with slope 4 and y-intercept -3.

1. The problem asks to find the table of variation of the function $g(x)$ based on the images provided. 2. Since the images are not accessible, I will explain how to find a table o

1. The problem asks to redo number 7 with a table of variation and explain step by step. 2. First, identify the function from problem 7 (assuming it is $f(x)$).

1. **Problem statement:** Consider the function $f$ defined on $]0; +\infty[$ by $f(x) = 2x(1 - \ln x)$. We need to find the limits of $f$ as $x \to 0$ and $x \to +\infty$. 2. **Li

1. **Problem statement:** We consider the function $f$ defined on $]0; +\infty[$ by $f(x) = 2x(1 - \ln x)$. We analyze its derivative, limits, intersections, tangents, inverse func

1. **Problem statement:** We have a function $f$ with curve $(C)$, tangent $(T)$ at point $A$, and vertical line $(d)$ with equation $x=1$. Point $B(1; 2e - 2)$ lies on $(d)$ and $

1. **Problem statement:** We have the function $f(x) = 2x(1 - \ln x)$ defined on $]0, +\infty[$. We want to find the limits of its derivative $f'(x)$ as $x \to 0$ and $x \to +\inft

1. **State the problem:** We need to find the derivative of the function $f(x) = 2x(1 - \ln x)$. 2. **Recall the formula:** To differentiate a product of two functions, use the pro

1. **State the problem:** Find the area between the curves $y = x e^{x^2}$ and $y = x^2$ for $x \geq 0$ from $x=0$ to $x=2$. 2. **Formula used:** The area between two curves $y=f(x

1. **Stating the problem:** We are given the function $$f(x) = (2 - x^2)(\ln x + 1)$$ and asked to analyze it according to the following points: 2. **Domain (תחום הגדרה):**

1. **State the problem:** Evaluate the integral $$\int \frac{\sec \theta}{\cot \theta} \, d\theta$$. 2. **Rewrite the integrand using trigonometric identities:** Recall that $$\cot

1. **State the problem:** Find the area bounded by the curve $y = 1 - x^2$ and the line $y = 0$. 2. **Identify the region:** The parabola $y = 1 - x^2$ opens downward and intersect

1. **Problem:** Find the limit $$\lim_{x \to 0} x \lfloor \cos x \rfloor$$ where $$\lfloor . \rfloor$$ is the greatest integer function. 2. **Recall:** The greatest integer functio

1. The problem is to evaluate the double integral $$\int_{-3}^0 \int_1^3 (10 + xy \sin(x^2 - y^2)) \, dy \, dx.$$\n\n2. The integral is over $x$ from $-3$ to $0$ and $y$ from $1$ t

1. **State the problem:** We need to form a differential equation from the given implicit relation $$y^2 = (x+c)^3$$ where $c$ is a constant. 2. **Rewrite the equation:** The equat

1. **Problem Statement:** Solve the alternating series \(\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n(n+1)} = 1\). 2. **Definition:** An alternating series is a series whose terms alterna