∫ calculus

1. The problem is to find the derivative of the function $x^2$ with respect to $x$. 2. The formula for the derivative of a power function $x^n$ is given by the power rule:

1. The problem is to determine whether the infinite series $$\sum_{n=1}^{\infty} \frac{5n^2 + 7}{n^3 + 9}$$ converges or diverges. 2. To analyze convergence, we use the Comparison

1. Problem: Beregn det ubestemte integralet $$\int x \sin(x^2) \, dx$$. 2. Formel: Vi bruker substitusjonsmetoden for integrasjon. La $$u = x^2$$, da er $$du = 2x \, dx$$ eller $$\

1. **State the problem:** We need to find the derivative $\frac{dy}{dx}$ of the function $$y = (4x^2 + 3)^3 e^{(3x^2 + 2)^2}.$$\n\n2. **Identify the formula and rules:** This is a

1. **Problem Statement:** Find the derivative of a function using the rules of derivatives. 2. **Derivative Rules:** The most common rules are:

1. **Stating the problem:** Given the function $$Y = \frac{6}{5} x^5 - \frac{2}{7} x^6$$, we want to analyze it, find its critical points, and understand its behavior.

1. The problem asks to find the Laplace transform of the function $$f(t) = 4t^3 + 7t^3 e^{12t}$$. 2. Recall the Laplace transform formulas:

1. **State the problem:** We want to find the integral $$\int (\tan x + \sec x)^2 \, dx$$. 2. **Use the formula:** Expand the square using algebraic identity $$(a+b)^2 = a^2 + 2ab

1. نبدأ بكتابة المتراجحة المعطاة: $$\ln(\ln x)' > 0$$. 2. نوجد مشتقة الدالة $$y = \ln(\ln x)$$ باستخدام قاعدة السلسلة:

1. **State the problem:** We need to find the indefinite integral $$\int \left(e^{3x} + \frac{2}{1-x}\right) dx.$$\n\n2. **Recall the integral rules:**\n- The integral of an expone

1. **State the problem:** We need to find the indefinite integral $$\int \left(e^{3x} + \frac{2}{1-x}\right) dx.$$\n\n2. **Recall the integral rules:**\n- The integral of a sum is

1. **Problem Statement:** Determine the intervals where the function $f(x) = x^3 - 6x^2 + 15$ is increasing or decreasing using the first derivative test. 2. **Formula and Rules:**

1. The problem is to simplify the inequality $\frac{d}{dx} \ln(\ln x) > 0$. 2. Recall the derivative rule for a composite function: if $y = \ln(u)$, then $\frac{dy}{dx} = \frac{1}{

1. **بيان المسألة:** لدينا الدالة $f(x) = \frac{2x^2}{\sqrt{x^2 + 8}}$ معرفة على $\mathbb{R}^+$.

1. The problem is to find the solution set of the inequality $\frac{d}{dx} \ln(\ln x) > 0$ without directly using the derivative. 2. First, recall the function: $y = \ln(\ln x)$.

1. **State the problem:** We need to find the solution set for the inequality $$\frac{d}{dx} \ln(\ln x) > 0$$. 2. **Find the derivative:** Using the chain rule, the derivative of $

1. **بيان المسألة:** لدينا الدالة $f$ المعرفة على $\mathbb{R}^+$ كما يلي: $$f(x) = \frac{2x^2}{\sqrt{x^3 + 8}}$$

1. Problem: Calculate the indefinite integral $$\int \frac{7x - 2}{x^2 - 1} dx$$. 2. Formula and rules: Use partial fraction decomposition since the denominator factors as $$x^2 -

1. **Stating the problem:** You asked about the antiderivative, integration, their graphical representation, real-life examples, and important formulas of integration. 2. **What is

1. **State the problem:** Given the function $y=3\sqrt{5x^2}$, find the derivative $y'$. 2. **Recall the formula:** The derivative of $y = a\sqrt{u}$ where $a$ is a constant and $u

1. The problem asks to sketch a function $g(x)$ with specific limit behaviors and discontinuities. 2. Let's analyze each condition and describe the graph behavior: