∫ calculus

1. نبدأ بحساب مشتقة الدالة $G(x) = x e^2 \sin\left(\frac{1}{x e^2}\right)$. 2. نلاحظ أن الدالة هي حاصل ضرب بين دالتين: $u = x e^2$ و $v = \sin\left(\frac{1}{x e^2}\right)$.

1. نبدأ بكتابة المسألة: نريد حساب نهاية الدالة $$\lim_{x \to 0} \frac{x^2 \sin\left(\frac{1}{x^2}\right)}{x}$$. 2. نبسط التعبير داخل النهاية: \n\n$$\frac{x^2 \sin\left(\frac{1}{x^2

1. نبدأ بكتابة المشكلة: نريد حساب نهاية الدالة $$\lim_{x \to 0} \frac{d}{dC} \sin\left(\frac{1}{x^2 e}\right)$$ حيث $C$ هو المتغير الذي نشتق بالنسبة له. 2. نلاحظ أن الدالة المعطاة

1. **Problem statement:** Find the limit as $x \to 0$ of $$\sqrt{3x + 5} - \sqrt{3x - 3}.$$\n\n2. **Formula and approach:** When dealing with limits involving differences of square

1. Задачата е да намерим производната на функцията $$f(x) = \frac{\sin x}{x} + (x + 2)^{3x - 1}$$. 2. За да намерим производната на сумата, използваме правилото, че производната на

1. **State the problem:** Evaluate the integral $$\int \sqrt{9 - x^2} \, dx$$. 2. **Identify the formula and substitution:** This integral is of the form $$\int \sqrt{a^2 - x^2} \,

1. **State the problem:** We are asked to find various limits and function values of the function $h(x)$ at points $x = -3$, $x = 0$, and as $x \to \infty$.\n\n2. **Recall limit de

1. **State the problem:** We are given a piecewise graph of a function $f$ and need to find the values of various limits and function values at $x=2$ and $x=4$. 2. **Recall limit d

1. **Problem:** Find the limit of the sequence $\left\{(n^2 + 2)^{1/n}\right\}_{n=1}^\infty$. 2. **Formula and rules:** For sequences of the form $a_n^{1/n}$, the limit can often b

1. The problem asks to find the limits and function values for the function $f$ at specific points based on the graph description. 2. Recall the definitions:

1. **Problem statement:** (a) Differentiate $y = \frac{\cos x}{\sin x}$ and show that $\frac{dy}{dx} = -\csc^2 x$.

1. **State the problem:** Calculate the definite integral $$\int_{-3}^{2} (\sin x + x^{-2} + \cos x) \, dx$$. 2. **Recall the integral formulas:**

1. **Problem statement:** Given the function $F(x)=3x^5-20x^3$, find its domain, parity, limits at domain boundaries, asymptotes, first and second derivatives with sign tables, int

1. **Problem:** Evaluate the integral $$\int \ln x \, dx$$ using integration by parts. 2. **Formula:** Integration by parts states:

1. **Problem a:** Evaluate $$\lim_{x \to \infty} \frac{x^4 - 2x^3 - 1}{x^2 - x}$$ 2. **Step 1:** Identify the highest powers of $x$ in numerator and denominator.

1. The problem is to find the integral of $\sinh x$ with respect to $x$, given as $\int \sinh x \, dx$. The statement also shows the result $\cosh x + C$, where $C$ is the constant

1. **State the problem:** Evaluate the integral $$\int \frac{dx}{4x^2 + 20x + 26}$$. 2. **Complete the square:** The quadratic in the denominator is $$4x^2 + 20x + 26$$.

1. **Problem:** Find the derivative of $$y = x \sqrt{\frac{x^2 - 1}{x^2 + 2x + 1}}$$. 2. **Step 1: Simplify the expression inside the square root.**

1. The problem is to determine the intervals where a function is increasing or decreasing. 2. To find these intervals, we use the first derivative test. The function is increasing

1. **State the problem:** Evaluate the integral $$\int_0^{\infty} \frac{x \ln x}{(1+x^2)^2} \, dx.$$\n\n2. **Recall the formula and approach:** This integral involves a logarithm a

1. **State the problem:** Find the equation of the tangent line to the function $h(x) = 3x^2 - 5$ at the point $(-1, 2)$. 2. **Recall the formula:** The equation of the tangent lin