∫ calculus

1. **Problem Statement:** Find the limit of a function as $x$ approaches infinity for various types of functions. 2. **General Idea:** The limit at infinity describes the behavior

1. لنبدأ بتوضيح المشكلة: تريد معرفة كيفية حساب مشتقة تركيب دالتين. 2. قاعدة مشتقة التركيب (قاعدة السلسلة) تقول: إذا كانت الدالة $h(x) = f(g(x))$، فإن مشتقتها هي

1. **Problem Statement:** Evaluate the integrals from 11 to 15 using integration techniques. 2. **Integral 11: \(\int \cos \sqrt{x} \, dx\)**

1. **Problem statement:** Differentiate $$\arctan \frac{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}$$ with respect to $$\arccos x^{2}$$. 2. **Rewrite the function

1. **Problem Statement:** We want to find the limit $$\lim_{x \to 4} \frac{x^3 - 2x^2 - 9x + 4}{3x^2 - 4x}$$

1. The problem is to determine if there is a comprehensive list of formulas covering integration. 2. Integration formulas include basic rules such as the power rule, sum rule, and

1. **Problem statement:** Find the volume of the solid bounded between the planes perpendicular to the x-axis at $x=0$ and $x=1$, where the cross sections perpendicular to the x-ax

1. **بيان المسألة:** ندرس الدالة $$f(x) = \frac{\sqrt{x^2 + x}}{x-1}$$ ونحسب النهايات التالية: - $$\lim_{x \to +\infty} f(x)$$

1. **State the problem:** Find the limit $$\lim_{x \to -\infty} 2\sqrt{x^2+3} - x - 3.$$\n\n2. **Recall the formula and rules:** When $x \to -\infty$, $x^2$ dominates inside the sq

1. **State the problem:** Differentiate the function $$y = 11 e^{0.05x} + 4$$ with respect to $$x$$. 2. **Recall the differentiation rule for exponential functions:** The derivativ

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{\sin(\pi \cos^2 x)}{x^2}.$$\n\n2. **Recall relevant formulas and rules:**\n- The limit $$\lim_{x \to 0} \frac{\sin x

1. **Problem Statement:** Determine whether the series $$\sum_{n=1}^{\infty} \left(\sqrt{n^{2}+1} - n\right)$$ converges. 2. **Recall the Test for Convergence:** For a series $$\su

1. **State the problem:** Differentiate the function $y = \sin^2(x^2)$ with respect to $x$. 2. **Recall the formula:** To differentiate a composite function like $\sin^2(u)$ where

1. Problem statement: Find the area bounded by the given curves for problems 1 through 5. 2. Problem 1: Find the area bounded by $y = x^2$ and $y = x$.

1. Problem 1: Find the area bounded by the curves $y = x^2$ and $y = x$. 2. Formula used: Area between curves is $$\int_a^b (\text{top}-\text{bottom})\,dx$$.

Problem: Evaluate the following seven definite integrals. 1. Problem: Compute $$\int_0^1 (3 - 2x)\,dx$$.

1. The problem is to understand whether definite integrals were used in previous calculations. 2. A definite integral is an integral with upper and lower limits, written as $$\int_

1. Calculate $$\int_0^1 (3 - 2x) \, dx$$ $$= \left[3x - x^2\right]_0^1 = (3(1) - 1^2) - (0) = 3 - 1 = 2$$

1. **Problem:** Calculate the definite integral $$\int_0^1 (3 - 2x) \, dx$$ 2. **Formula:** The definite integral of a function $f(x)$ from $a$ to $b$ is given by $$\int_a^b f(x) \

1. **State the problem:** We need to express the integrand $$\frac{x^3}{(x+1)^2}$$ in partial fractions and then evaluate the integral $$\int \frac{x^3}{(x+1)^2} \, dx$$. 2. **Set

1. مسئله: تابع قطعه‌ای $f(x)$ به صورت زیر تعریف شده است: $$f(x) = \begin{cases} 6ax^2 + 1 & x > 2 \\ 3 & x = 2 \\ \frac{\sqrt{x^2 - 4x + 4}}{-2x + 4} & x < 2 \end{cases}$$