∫ calculus

1. **Stating the problem:** We want to analyze and graph the function $$f(x) = 2x + \ln(x^2 - 3)$$ with the given domain and properties.

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{e^{2x}}{x^{1/2}(x^2+5)^{1/4}}$$. 2. **Rewrite the function:** To differentiate, rewrite the function as

1. **State the problem:** Find the limit as $x \to +\infty$ of the expression $$\frac{\sqrt{x} + \sqrt{x} + \sqrt{x}}{\sqrt{x} + 1}.$$\n\n2. **Simplify the numerator:** The numerat

1. **State the problem:** Find the limit $$\lim_{x \to +\infty} \frac{\sqrt{x} + \sqrt{x + \sqrt{x}}}{\sqrt{x} + 1}$$ as $x$ approaches positive infinity. 2. **Recall the formula a

1. Problem: Find the first derivative $\frac{dy}{dx}$ for $y=2^x \sin^{-1} x$. Use the product rule since $y$ is a product of two functions. 2. Formula: Product rule states $\frac{

1. **State the problem:** Find the limits using the definition of the derivative (first principle): $$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

1. **Stating the problem:** We want to analyze and graph the function $$f(x) = \frac{x^2}{(x-1)^2}$$ with the given properties about domain, monotonicity, local minima, concavity,

1. **State the problem:** We need to evaluate the integral $$\int \left( \frac{3}{\sqrt{1 - x^2}} + \frac{4}{\sqrt{1 + x^2}} + \frac{5x}{\sqrt{x^2 - 1}} \right) dx$$

1. **Problem Statement:** Given the function $f(x) = e^x$, show that its derivative $f'(x) = e^x$. 2. **Formula Used:** The derivative of the exponential function $e^x$ with respec

1. We are asked to evaluate the integral $$2 \int \frac{\sin x}{3 + \cos 2x} \, dx$$. 2. Recall the double-angle identity: $$\cos 2x = 2\cos^2 x - 1$$.

1. **Stating the problem:** We need to evaluate the expression

1. **State the problem:** We want to evaluate the expression

1. **State the problem:** We need to find the right-hand limit of the function $f(x)$ as $x$ approaches $-4$ from the right, denoted as $\lim_{x \to -4^+} f(x)$. 2. **Understand th

1. **State the problem:** We need to find all values of $a$ in the interval $-9 < x < 9$ such that $$\lim_{x \to a} f(x) = -3.$$ This means the function $f(x)$ approaches the value

1. **State the problem:** We need to find the limit $$\lim_{x \to -\infty} \frac{2x - 4x^3}{6x^2 + e^{-x}}.$$\n\n2. **Identify dominant terms:** As $x \to -\infty$, the term $-4x^3

1. **Problem:** Differentiate the function $$h(x) = (2x + 5)^7 (3x^4 - 8)^5$$. 2. **Formula and rules:** To differentiate a product of two functions, use the product rule:

1. المشكلة: نريد معرفة كيفية اشتقاق الدوال. 2. الاشتقاق هو عملية حساب معدل التغير اللحظي للدالة، أو ميل المماس لمنحنى الدالة عند نقطة معينة.

1. **Problem statement:** Find the monotony intervals of the function $$f(x) = (x + 13) |x + 13|$$. 2. **Rewrite the function:** The absolute value function splits into cases:

1. **State the problem:** Check the differentiability of the function $$f(x) = 1 - 3\sqrt{(x-1)^2}$$. 2. **Rewrite the function:** Note that $$\sqrt{(x-1)^2} = |x-1|$$, so the func

1. **Stating the problem:** We need to check the differentiability of the function $$f(x) = 1 - 3\sqrt{(x-1)^2}$$. 2. **Rewrite the function:** Note that $$\sqrt{(x-1)^2} = |x-1|$$

1. **Problem Statement:** Calculate the double integral $$\iint_D 3(x+1)^2 (y+2)^2 \, dx \, dy$$ over the square $D$ with vertices $(1,0)$, $(0,1)$, $(0,-1)$, and $(-1,0)$.