∫ calculus

1. **Problem statement:** Evaluate the double integral $$\iint_R (x+1) \, dA$$ where $R$ is the region bounded by $y=1$ and $y=x^2$ for $x$ in $[-2,2]$. 2. **Set up the integral:**

1. **Tính tích phân**: 2. a/ Tính $I_1 = \int (\tan x + 3e^{-2x}) \, dx$.

1. We are asked to solve the integral \(\int x^5 \sqrt[3]{2 - x^3} \, dx\). 2. To solve this integral, we use substitution. Let \(u = 2 - x^3\).

1. **State the problem:** Find the limit $$\lim_{x \to 2} \left(x^3 - 4x + 1\right)^{\frac{1}{x^2 - 4}}.$$\n\n2. **Identify the form:** Substitute $x=2$ directly:

1. **State the problem:** Find the equation of the tangent line to the function $f(x) = 3 - 2x$ at the point $(-1, 5)$. 2. **Recall the formula:** The equation of the tangent line

1. **State the problem:** Find the second derivative $\frac{d^2y}{dx^2}$ given the implicit differentiation result for $\frac{dy}{dx} = \frac{2y - x^2}{y^2 - 2x}$. 2. **Recall the

1. **State the problem:** Differentiate implicitly the equation $$x^3 + y^3 = 6xy$$ with respect to $x$ to find $\frac{dy}{dx}$. 2. **Recall the rules:**

1. The problem is to understand the course outline for a calculus or advanced mathematics course covering sequences, series, power series, partial derivatives, and multiple integra

1. **State the problem:** Differentiate the function implicitly given by the equation $6xy$ with respect to $x$. 2. **Recall the product rule:** When differentiating a product of t

1. The problem asks to find the derivative of the function $6xy$ with respect to both $x$ and $y$. 2. Since $6xy$ is a product of two variables, we use partial derivatives.

1. **Problem:** Determine if the function $f(x)=x^3+x^2-2$ is continuous at $x=1$. 2. **Formula and rule:** A function is continuous at $x=a$ if $\lim_{x \to a} f(x) = f(a)$.

1. **State the problem:** Given the function $y=2x^3 + 7x^2 + 4x - 3$, we need to find the stationary points, the point of inflection, and sketch the curve. 2. **Recall formulas an

1. **State the problem:** Evaluate the improper integral $$\int_{-1}^1 \frac{1}{(1+x)^{2/3}} \, dx.$$\n\n2. **Identify the issue:** The integrand has a potential problem at $x = -1

1. **Stating the problem:** We want to find the critical points of a function, which are points where the function's slope is zero or undefined. 2. **Formula and rules:** Critical

1. **State the problem:** Find the minima and maxima of the function $$y=(x-4)^4(x+3)^3$$. 2. **Formula and rules:** To find minima and maxima, we first find the critical points by

1. **State the problem:** We need to evaluate the double integral $$\int_0^1 \int_0^1 v (u + v^2)^4 \, du \, dv.$$\n\n2. **Understand the integral:** The integral is over $u$ from

1. **Problem Statement:** Fill in the table with the appropriate values of $\lim_{x \to c} f(x)$ and $f(c)$ based on the given graph.

1. The problem is to understand the concept of differentiation and how to find the derivative of a function. 2. Differentiation is the process of finding the rate at which a functi

1. **Stating the problem:** Evaluate the expression involving an integral and a limit:

1. **State the problem:** Find the derivative of a function $f(x)$. Since the function is not specified, let's consider a general approach. 2. **Formula used:** The derivative of a

1. Problem: Calculate the integral $$\int \frac{\cos 5x}{3 + \sin 5x} \, dx$$. 2. Formula and rules: Use substitution for integrals involving trigonometric functions. Let $$u = 3 +