∫ calculus

1. **Problem Statement:** Find the antiderivative (indefinite integral) of the function $f(x) = 5x^2$. 2. **Formula Used:** The antiderivative of a power function $x^n$ is given by

1. **State the problem:** Find the area between the curves $y=x$, $y=4x$, and $y=-x+2$. 2. **Find the points of intersection:**

1. **State the problem:** We need to find the limit $$\lim_{x \to 1} \frac{x^{2} + 5x + 4}{x^{2} - 2x - 3}$$. 2. **Recall the formula and rules:** To find limits of rational functi

1. **State the problem:** Evaluate the limit $$\lim_{x \to 100} \frac{\sqrt{x} - 10}{x - 100}$$. 2. **Recall the formula and rules:** Direct substitution gives $$\frac{\sqrt{100} -

1. **Problem Statement:** Find the derivative with respect to $x$ of the integral $$\frac{d}{dx} \left( \int_{x^2}^x t^2 \, dt \right).$$ 2. **Recall the Fundamental Theorem of Cal

1. **State the problem:** We need to evaluate the integral $$\int \frac{u}{u^2 - 2u + 1} \, du$$. 2. **Simplify the denominator:** Notice that $$u^2 - 2u + 1 = (u - 1)^2$$.

1. **State the problem:** We need to find the derivative of the function $$f(x) = \frac{1}{\sqrt{x^6 + 2x + 1}}.$$\n\n2. **Rewrite the function:** To differentiate easily, rewrite

1. The problem is to evaluate the definite integral $$\int_1^{e-1} \ln(n+1) \, dn$$. 2. We use integration by parts formula: $$\int u \, dv = uv - \int v \, du$$.

1. **State the problem:** We are given the total cost function $$C = 1500 + 2x - 5x^2$$ and the total revenue function $$R = x^2 - 2x$$. We need to find the rate of change of cost

1. **State the problem:** Find the limit $$\lim_{n \to \infty} \frac{n \log n}{n^2 - n}$$. 2. **Recall the formula and rules:** When evaluating limits involving infinity, compare t

1. Let's start by stating the problem: understanding the concept of limits in calculus. 2. A limit describes the value that a function $f(x)$ approaches as the input $x$ approaches

1. The problem is to find $\frac{dy}{dx}$ by implicit differentiation for the equation $$x^2 + y^2 = 100.$$\n\n2. The formula used is implicit differentiation, which means we diffe

1. **State the problem:** We need to find the left-hand limit of the piecewise function $$f(x)$$ as $$x$$ approaches 3 from the left, i.e., $$\lim_{x \to 3^-} f(x)$$. 2. **Identify

1. The problem asks to find the right-hand limit of the function $f(x)$ as $x$ approaches 0, denoted as $\lim_{x \to 0^+} f(x)$. 2. The right-hand limit means we look at values of

1. **Problem:** Find the limit $$\lim_{t \to 0} t \tan t$$ using L'Hospital's rule. 2. **Check form:** As $t \to 0$, $t \to 0$ and $\tan t \to 0$, so the product $t \tan t \to 0$.

1. **Problem Statement:** You want a comprehensive, easy-to-understand guide covering all key calculus topics in your syllabus to prepare for your exam in two days. 2. **Framework

1. **Problem:** Evaluate the limit $$\lim_{x \to 2} f(x)$$ where $$f(x) = \frac{x^3 - 4x^2 + x + 6}{x^3 - 6x^2 + 11x - 6}$$. 2. **Formula and rules:** To find the limit of a ration

1. **State the problem:** Calculate the integral $$\int \frac{1 + x + x^2}{x^4} \, dx$$. 2. **Rewrite the integrand:** Divide each term in the numerator by $x^4$:

1. Problem: Find the area bounded by the curves $y^2 = 9x$ and $y = 3x$. 2. Formula and rules: The area between two curves $y=f(x)$ and $y=g(x)$ from $x=a$ to $x=b$ is given by $$\

1. مسئله را بیان می‌کنیم: باید تابع $f(x)$ را پیدا کنیم که در معادله $$\int \frac{(\sqrt{x} - 1)(x + \sqrt{x})}{x^2} \, dx = \frac{1}{\sqrt{x}} f(x) + C$$

1. مسئله: یافتن تغییر متغیر مناسب برای انتگرال $$\int \frac{dx}{\sqrt{x} + \sqrt[3]{x}}$$ است. 2. برای ساده کردن عبارت مخرج، باید تغییر متغیری انتخاب کنیم که توان‌های کسری $\frac{1