∫ calculus

1. Let's solve an example integral: $$\int (3x^2 + 2x + 1) \, dx$$. 2. The formula for integrating a power function is $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ where $n \neq -1

1. The problem gives the derivative of a function as $f'(x) = (x^2 + 1) \sin(3x - 1)$ and asks about intervals where the function is increasing or decreasing. 2. Recall that a func

1. **Problem Statement:** We are given the function $$f(x) = \frac{a}{x^2 + b^2}$$ where $a, b > 0$. We need to find the critical points and possible points of inflection in terms

1. **State the problem:** Given the function $f(x) = 2x - 1$, find the first and second derivatives, identify the correct graph for $f$ and $f''$, and determine intervals where $f$

1. The problem states that we have a function $y = f(x)$ and another function defined as $g(x) = x + f(x)$. We are asked to find the graph of the derivative $y = g'(x)$. 2. Recall

1. **State the problem:** We need to find $\frac{dy}{dx}$ at the point $(2,0)$ for the implicit equation $$\tan^{-1}(x^2 y) = 4x + xy - 8.$$\n\n2. **Recall the formula and rules:**

1. We are given a function $h(x) = f\left(\frac{f(x) + 2}{3}\right)$ and asked to find $h'(1)$.\n\n2. To find $h'(x)$, we use the chain rule. Let $u = \frac{f(x) + 2}{3}$. Then $h(

1. **Problem statement:** Show that if $y = e^x$, then $$y_{n+1} - 2y_n - 2xy_n' + n(n - 1)y_{n-1} = 0.$$ 2. **Recall definitions and formulas:** Here, $y_n$ denotes the $n$th deri

1. The problem asks to find the derivative of the function $f(x) = 2\sqrt{x}$ at $x=4$. 2. Recall the derivative rule for $\sqrt{x}$: if $f(x) = \sqrt{x} = x^{1/2}$, then

1. **State the problem:** We need to prove that the equation $g(x) = x^2 + 2\ln x = 0$ has exactly one solution for $x > 0$. 2. **Recall the domain and function:** The function $g(

1. **State the problem:** Find the limit $$\lim_{x \to \pi} \frac{f'(x) - f'(\pi)}{\pi - x}$$ where $$f(x) = e^x \cos x + \sin x$$.

1. لنبدأ بتحديد الدالة المعطاة: $$g(x) = x^2 + 2\ln(x)$$. 2. نريد معرفة إذا كانت الصفر قيمة ممنوعة في مشتقة الدالة.

1. **State the problem:** We need to find the derivative of the function $$g(x) = x \cdot 2 + 2 \ln x = 2x + 2 \ln x$$ and then analyze the intervals where the function is increasi

1. **State the problem:** Evaluate the integral $$\int_0^1 (x^2 \sin y - x \cos y) \, dx$$ where $y$ is treated as a constant with respect to $x$. 2. **Recall the integral rules:**

1. **State the problem:** We are given the function $f(x,y) = x^2 y - 3xy^3$ and asked to evaluate the definite integral $$\int_1^2 f(x,y) \, dx$$ treating $y$ as a constant. 2. **

1. نبدأ ببيان المسألة: لدينا مساحة مستطيل تساوي $5 \times 6 = 30$. 2. المطلوب هو حساب مساحة المنحنى المعطى بالتكامل:

1. **State the problem:** Find the limit $$\lim_{x \to 1} 4^{\frac{2x - 2}{x}}$$. 2. **Recall the limit and exponential rules:** The limit of an exponential function $$a^{f(x)}$$ a

1. **Problem:** Evaluate the integral $$\int \cot^3(5x) \csc^5(5x) \, dx$$. 2. **Formula and rules:** Recall that $$\cot x = \frac{\cos x}{\sin x}$$ and $$\csc x = \frac{1}{\sin x}

1. **State the problem:** We want to evaluate the integral $$\int \frac{3x^3 - x^2 + 2x - 4}{\sqrt{x^2 - 3x + 2}} \, dx$$

1. We are asked to find the limit: $$\lim_{x \to 0} \frac{\cos 4x \cdot \sin 3x}{5x}$$ 2. Recall the important limit rules:

1. Problem: Calculate the indefinite integral $$\int (x + x^4 + 2x^5) \, dx$$. 2. Formula: The integral of a sum is the sum of the integrals, and the integral of $$x^n$$ is $$\frac