∫ calculus

1. The problem is to understand whether definite integrals were used in previous calculations. 2. A definite integral is an integral with upper and lower limits, written as $$\int_

1. Calculate $$\int_0^1 (3 - 2x) \, dx$$ $$= \left[3x - x^2\right]_0^1 = (3(1) - 1^2) - (0) = 3 - 1 = 2$$

1. **Problem:** Calculate the definite integral $$\int_0^1 (3 - 2x) \, dx$$ 2. **Formula:** The definite integral of a function $f(x)$ from $a$ to $b$ is given by $$\int_a^b f(x) \

1. **State the problem:** We need to express the integrand $$\frac{x^3}{(x+1)^2}$$ in partial fractions and then evaluate the integral $$\int \frac{x^3}{(x+1)^2} \, dx$$. 2. **Set

1. مسئله: تابع قطعه‌ای $f(x)$ به صورت زیر تعریف شده است: $$f(x) = \begin{cases} 6ax^2 + 1 & x > 2 \\ 3 & x = 2 \\ \frac{\sqrt{x^2 - 4x + 4}}{-2x + 4} & x < 2 \end{cases}$$

1. **Problem 1:** Evaluate $$\int \frac{x^3}{(x+1)^2} \, dx$$ by expressing the integrand in partial fractions. 2. **Problem 2:** Evaluate $$\int \frac{11x - 66}{x^2 - x - 2} \, dx

1. مسئله: تابع بخش‌بندی شده $f(x)$ به صورت زیر تعریف شده است: $$f(x) = \begin{cases} 6ax^2 + 1 & x > 2 \\ 3 & x = 2 \\ \frac{\sqrt{x^2 - 4x + 4}}{-2x + 4} & x < 2 \end{cases}$$

1. مسئله را بیان می‌کنیم: می‌خواهیم حد $$\lim_{x \to 2} \frac{\sqrt{2x - 2}}{x^2 - 3x + 2}$$ را بیابیم. 2. ابتدا صورت و مخرج را بررسی می‌کنیم. اگر مستقیماً $$x=2$$ را جایگذاری کنیم

1. مسئله: باید حد $$\lim_{x \to 2} \frac{\sqrt{2x - 2}}{x^3 - 3x + 2}$$ را محاسبه کنیم. 2. ابتدا مقدار صورت و مخرج را در $$x=2$$ جایگذاری می‌کنیم:

1. مسئله: مقدار عبارت $$2 \lim_{x \to 2} f(x) - \left[ \lim_{x \to 2} f(x) \right]$$ را بیابید که در آن نماد \( [\cdot] \) نشان‌دهنده جزء صحیح است. 2. ابتدا باید مقدار $$\lim_{x \t

1. **Problem statement:** Find the limit $$\lim_{x \to 2} \frac{\sqrt{2x - 2}}{x^2 - 3x + 2}$$. 2. **Recall the formula and rules:** To find limits involving square roots and polyn

1. **State the problem:** Calculate the integral $$\int 1 \, dx$$. 2. **Formula used:** The integral of a constant $c$ with respect to $x$ is given by $$\int c \, dx = cx + C$$ whe

1. **State the problem:** Express the rational function $$\frac{19 - x}{(x - 7)(x + 4)}$$ as the sum of its partial fractions and then find the integral $$\int \frac{19 - x}{(x - 7

1. **State the problem:** Express the rational function $$\frac{19 - x}{(x - 7)(x + 4)}$$ as the sum of its partial fractions and then find the integral $$\int \frac{19 - x}{(x - 7

1. **State the problem:** Express the rational function $$\frac{19 - x}{(x - 7)(x + 4)}$$ as the sum of its partial fractions and then find the integral $$\int \frac{19 - x}{(x - 7

1. **State the problem:** Express the rational function $$\frac{19 - x}{(x - 7)(x + 4)}$$ as a sum of partial fractions and then find the integral $$\int \frac{19 - x}{(x - 7)(x +

1. Masalah: Hitung luas daerah di bawah kurva $f(x) = x^2 - 4$ dan di atas sumbu $x$ pada interval $[0,3]$. 2. Rumus yang digunakan adalah integral tentu dari fungsi tersebut pada

1. **Problem Statement:** Evaluate the integral $$\int x^{\mu^{x^r}} \, dx$$ where $\mu$ and $r$ are constants. 2. **Understanding the problem:** This integral involves a variable

1. **Stating the problem:** Hitung nilai dari integral tertentu $$\int_0^3 (2x^2 - 4x + 1) \, dx$$ dan tentukan juga integral tak tentu dari fungsi tersebut.

1. **State the problem:** Find the limit as $x$ approaches $-\infty$ of the function $$\frac{2x^4 + 4x^3 - 1}{4x^3 - 3x^5 + 5}.$$\n\n2. **Identify the highest powers:** The numerat

1. **State the problem:** Find the limit $$\lim_{x \to \frac{\pi}{\gamma}^+} \frac{1}{\cos x}$$ where $x$ approaches $\frac{\pi}{\gamma}$ from the right. 2. **Recall the behavior o