∫ calculus

1. **Problem statement:** Find the values of $x$ for which the derived function $f'(x) = 3x^2$ is zero, and prepare a gradient table to determine whether these points are maxima, m

1. **State the problem:** Evaluate the definite integral $$\int_2^4 \frac{x^2 + 2x + 4}{x^3 + 4x} \, dx.$$\n\n2. **Simplify the integrand:** Factor the denominator:\n$$x^3 + 4x = x

1. **Problem statement:** Given the graph of the derivative $y = f'(x)$ of a quartic function $y = f(x)$, analyze the following statements: a) $f(x)$ is decreasing on $(-\infty, -2

1. **State the problem:** Find the derivative $f'(x)$ of the function $f(x) = -4x^8$. 2. **Recall the power rule for derivatives:** If $f(x) = ax^n$, then $f'(x) = a n x^{n-1}$.

1. مسئله: محاسبه انتگرال تابع سینوس یعنی $$\int \sin(x) \, dx$$. 2. فرمول: انتگرال تابع سینوس برابر است با $$-\cos(x) + C$$ که در آن $$C$$ ثابت انتگرال است.

1. نبدأ بكتابة المعادلة المعطاة: $$f(x) = \sin x + \cos x$$. 2. المطلوب هو إيجاد نقاط تقاطع مع المحورين (نقاط إكس وواي) بالإضافة إلى معادلة الخط العمودي على المماس عند $$x=\pi$$.

1. **State the problem:** Find the possible values of $k$ such that the limit $\lim_{x \to k} f(x)$ exists for the piecewise function:

1. **State the problem:** Find the limit $$\lim_{x \to 5} (x^2 - 3x)$$. 2. **Recall the limit rule for polynomials:** The limit of a polynomial function as $x$ approaches a value i

1. **State the problem:** We want to find the double integral $$\iint_D y \, dA$$ where $D$ is the region enclosed by the $x$-axis, the line $y=\frac{1}{3}x$, and the parabola $y=\

1. **Stating the problem:** Find the limit $$\lim_{n \to \infty} \frac{1 + r + r^2 + \cdots + r^n}{n + r} - \frac{n}{r}$$ where $r$ is a constant. 2. **Recall the formula for the s

1. مسئله: مشتق جهتی تابع $f(x,y) = x^3 - 3x^2 y - y^3$ در نقطه $p(1,-2)$ و جهت $\mathbf{u} = \mathbf{i} - \mathbf{j}$ را بیابید. 2. فرمول: مشتق جهتی تابع $f$ در نقطه $p$ به سمت برد

1. **State the problem:** Find the limit $$\lim_{x \to x^+} \left( \frac{4x^2}{9} - x^2 \right)$$. 2. **Rewrite the expression:**

1. The problem is to find the derivative of the function $f(x) = 3x^2 + 5x - 7$. 2. The formula for the derivative of a polynomial function $f(x) = ax^n$ is $f'(x) = n \cdot a x^{n

1) Calculate the limit \(\lim_{x \to +\infty} \left[\sqrt{x^3 + 5} - x\sqrt{x}\right]\). Step 1. State the problem: Find \(\lim_{x \to +\infty} \left[\sqrt{x^3 + 5} - x\sqrt{x}\rig

1. مسئله: مشتق تابع $\ln(\sin x)$ را پیدا کنید. 2. فرمول مورد استفاده: مشتق تابع مرکب $\ln(u)$ برابر است با $\frac{1}{u} \cdot u'$ که در اینجا $u = \sin x$ است.

1. **State the problem:** Find the area of the region bounded by the curves $$y=\frac{1}{x+3}$$, $$y=(x+3)^2$$, and the vertical lines $$x=-\frac{5}{2}$$ and $$x=1$$. 2. **Set up t

1. **State the problem:** Find the surface area of the solid formed by rotating the curve $y = x^2 + 3$ for $1 < x < 5$ about the y-axis. 2. **Formula for surface area of revolutio

1. **Problem Statement:** Determine the following limits: a) $$\lim_{x \to +\infty} (3x^4 - 5x^2 + 8)$$

1. The problem is to find the derivative of the function $y = e^{x^x}$ with respect to $x$. 2. We use the chain rule for differentiation, which states that if $y = e^u$ where $u$ i

1. បញ្ហា: រកអានុគមន៍ច្រាសនៃអានុគមន៍ $f(x) = \frac{3x - 1}{x + 2}$។ 2. អានុគមន៍ច្រាស (derivative) នៃអនុគមន៍អនុគមន៍ភាគរយ $\frac{u(x)}{v(x)}$ គឺ $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{

1. **Problem Statement:** Evaluate the integral $$\iint_R e^{x^2 + y^2} \, dy \, dx$$ where $R$ is the semicircular region bounded by the $x$-axis and the curve $y = \sqrt{1 - x^2}