∫ calculus

1. **Problem a:** Evaluate $$\lim_{n \to \infty} \sqrt{4^n + 5^n}$$ - Step 1: Identify the dominant term inside the square root as $$n \to \infty$$.

1. نحدد المشكلة: نحسب النهاية $$\lim_{x \to \frac{\pi}{4}} \frac{\tan x - 1}{\sin x - \frac{\sqrt{2}}{2}}$$. 2. نعوض مباشرةً بـ $$x = \frac{\pi}{4}$$.

1. The problem is to find the derivative $y'(x)$ of the function $y(x) = x^x$. 2. To differentiate $y = x^x$, first rewrite it using logarithms:

1. **State the problem:** We are given the function $y = x^2 e^{2x}$ and need to find the first derivative $y'$, the second derivative $y''$, and then calculate the price elasticit

1. **State the problem:** Find the limit $$\lim_{x \to +\infty} \frac{3x^2 + 7}{5x - 1}$$.\n\n2. **Analyze the degrees:** The numerator is a polynomial of degree 2 ($3x^2 + 7$) and

1. **Problem statement:** Given the function $$P = 2Q^3 + 5Q^2 + \frac{2}{Q}$$, find its derivative with respect to $$Q$$, denoted as $$\frac{dP}{dQ}$$. 2. **Rewrite the function f

1. Problem: Evaluate the indefinite integral $$\int \sin(2x)\,dx$$. 2. Use the substitution rule: let $$u = 2x$$, then $$\frac{du}{dx} = 2$$ or $$dx = \frac{du}{2}$$.

1. **استخدام نظرية الشطيرة لإيجاد الحد** المشكلة: \(\lim_{x \to 0} (3 + x^2 \sin(\frac{1}{x}))\)

1. المشكلة: نجد قيمة \( \lim_{x \to 0^+} \frac{\sqrt{1 - \cos x}}{x} \).\n\n2. معطى: \( \lim_{x \to 0^+} \frac{1 - \cos x}{x^2} = \frac{1}{2} \).\n\n3. يمكننا كتابة التعبير المطلوب

1. نحدد طول قوس المنحنى $f(x) = x^3 + 2$ في الفترة $-1 \leq x \leq 1$ باستخدام $n=2$ قطع مستقيمة. 2. طول قوس المنحنى يقترب من مجموع أطوال القطع المستقيمة التي تصل بين النقاط على ال

1. Problem: Find $\frac{dy}{dx}$ if $x + y^4 = 10$ and $y \neq 0$. 2. Differentiate both sides with respect to $x$:

1. The problem is to evaluate the integral $$\int \frac{1}{x^2 (x+1)} \, dx.$$\n\n2. First, decompose the integrand into partial fractions. We want to express \( \frac{1}{x^2(x+1)}

1. **State the problem:** Find the limit $$L_{22} = \lim_{x \to 1} \frac{x^{25} - \sqrt{2x} - 1}{x - 1}.$$\n\n2. **Check substitution:** Substituting $x=1$ directly gives numerator

1. Find \( \lim_{x \to 7^-} \frac{x^2 + 5x - 19}{x + 7} \). Step 1: Substitute \( x = 7 \) into the expression.

1. We are asked to find the derivative of the function $$f(x) = \frac{2 \sin(x) - 7}{9x^9 - 3}$$ using the quotient rule. 2. Recall the quotient rule formula: if $$f(x) = \frac{u(x

1. The problem asks to find the derivative of the function $g(x) = \sin(x)$ and then evaluate this derivative at $x = 5$. 2. Recall that the derivative of $\sin(x)$ with respect to

1. **Problem statement:** Given functions $$u = \frac{2x - y}{2} x$$

1. Find \(\frac{dy}{dx}\) for the equation \(x^{3} + y^{3} = xy\). First, differentiate both sides with respect to \(x\), remembering that \(y\) is a function of \(x\).

1. **State the problem:** We need to find \(\frac{dy}{dx}\) by differentiating implicitly the equation \(x^3 + y^3 = xy\) with respect to \(x\). 2. **Differentiate each term:**

1. Differentiate implicitly the equation $x^3 + y^3 = xy$ with respect to $x$. 2. Use the product rule on the right-hand side and chain rule on $y^3$. Remember $y$ is a function of

1. **State the problem:** We have the sales function $$S(t) = \frac{125 t^2}{t^2 + 100}$$