∫ calculus

1. **State the problem:** We need to verify if the function $f(x) = 1 - \cos\left(x^{\frac{1}{2}}\right)$ satisfies the conditions of Rolle's theorem on the interval $\left(-\frac{

1. **State the problem:** We need to apply Rolle's Theorem to the function $f(x) = \sin x$ on the interval $(0, 4\pi)$.\n\n2. **Recall Rolle's Theorem:** If a function $f$ is conti

1. **Problem:** Find where the function $f(x) = x^3 - 1$ is increasing or decreasing using the Monotonicity Theorem. 2. **Formula and rules:** The Monotonicity Theorem states that

1. The problem asks to find the integral of the function $P(u) = \frac{4n-1}{x-1}$.\n\n2. The integral formula for a function of the form $\frac{a}{x-b}$ is $$\int \frac{a}{x-b} \,

1. **State the problem:** Find the limit $$\lim_{x \to -\infty} \frac{\sqrt{x^2 - x} + \sqrt{x^4 - x^6 + 1} - x^3}{x}.$$\n\n2. **Rewrite and analyze the expression:** The numerator

1. **State the problem:** Find the partial derivative of the function $F(x,y) = x \tan y + y^2 \cos x$ with respect to $x$, denoted as $F_x$. 2. **Recall the formula:** The partial

1. Problem: Find the derivative of the function $$f(x) = 3x^2 + 3\sqrt[3]{x} - \frac{2}{x^3}$$ 2. Formula and rules:

1. مسئله: محاسبه انتگرال $$\int \frac{1}{\sin^2 x} \, dx$$. 2. فرمول و قوانین مهم: تابع $$\frac{1}{\sin^2 x}$$ برابر است با $$\csc^2 x$$ که انتگرال آن شناخته شده است.

1. Problem statement: Compute the integral $\int \sin^2 x\, dx$. 2. Formula and rule: Use the power-reduction identity $\sin^2 x = \frac{1-\cos(2x)}{2}$ and the linearity of the in

1. مسئله: محاسبه انتگرال $$\int \sin^2 x \, dx$$ است. 2. فرمول مورد استفاده: برای انتگرال توانی از سینوس، از فرمول تبدیل استفاده می‌کنیم:

1. We are asked to find the limit $\lim_{x \to 1} f(4-2x)$.\n\n2. The limit expression is $\lim_{x \to 1} f(4-2x)$. To evaluate this, we substitute $x=1$ into the inner function: $

1. **Identify the derivatives:** (i) Given $y = (3x^2 + 1)^{\frac{3}{2}}$

1. **Problem statement:** Find the differential uncertainty (doubt) $\Delta y$ for the function $$y = \left(\frac{1}{a} + \frac{1}{\sqrt{b}}\right) \cdot \frac{2}{c + d^2}$$

1. **State the problem:** We need to evaluate the definite integral $$\int_0^3 \frac{dx}{x + \sqrt{9 - x^2}}.$$\n\n2. **Understand the integral:** The denominator is $x + \sqrt{9 -

1. Masalani bayon qilamiz: $\lim_{x \to 0} \frac{\cos 4x - \cos^3 4x}{4x^2}$ ni topish kerak. 2. Limitni hisoblash uchun $\cos 4x$ ning $x$ yaqinida Maclaurin qatorini ishlatamiz:

1. The problem is to find the limit: $$\lim_{x \to 3} \frac{x^2 + x - 12}{\sqrt{5x + 1} - 4}$$. 2. First, substitute $x = 3$ directly to check if the expression is defined:

1. **State the problem:** We want to evaluate the double integral $$\iint_R \frac{-y}{x^2 y^2 + 1} \, dA$$ where the region $$R$$ is the unit square defined by $$0 \leq x \leq 1$$

1. Diketahui fungsi pertama: $$y = 3x^2 (5x - 3)^4$$ 2. Kita akan menggunakan rumus integral parsial: $$\int u \, dv = uv - \int v \, du$$

1. **State the problem:** We have a curve $C$ defined by the equation $$x = \frac{2}{3} \sin(3y + \frac{\pi}{4})$$ with domain $$\frac{\pi}{12} < y < \frac{3\pi}{4}$$.

1. **Problem statement:** Find the area of the basin's horizontal cross-section at its widest point, the volume of water needed to fill the basin, and the lateral surface area gene

1. مسئله: مشتق توابع داده شده را بیابید. 2. فرمول‌ها و قواعد مهم: