∫ calculus

1. **State the problem:** Evaluate the integral $$\int (2x^2)(\sec^2 x)(\tan x) \, dx$$. 2. **Recall relevant formulas and rules:**

1. **State the problem:** We need to find the integral $$\int (\sec(x))^3 \, dx$$. 2. **Recall the formula and rules:** For integrals involving powers of secant, a useful approach

1. **State the problem:** Find the surface area of the solid formed by rotating the curve $y=2x^2$ for $0 \leq x \leq 1$ about the y-axis. 2. **Formula for surface area when rotati

1. **Problem:** Given $z = \tan^{-1}\left(\frac{x}{y}\right)$, find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$.\n\n2. **Formula and rules:** The derivative

1. **Problem:** Find the derivative of the function $$f(x) = \frac{1}{(1 - x^2)^{3/2}}$$. 2. **Formula and rules:** Use the chain rule and power rule. For $$f(x) = (g(x))^n$$, $$f'

1. **Problem statement:** Find the limit $$\lim_{x \to 2} \frac{x^4 - 8x}{\frac{1}{4}x^3 + 5x - 6x^2}$$ using L'Hopital's rule. 2. **Check the form:** Substitute $x=2$:

1. Problem: Find $\frac{dy}{dx}$ if $y = \ln 5x$. 2. Formula: The derivative of $y = \ln u$ with respect to $x$ is $\frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx}$.

1. **Problem Statement:** Find the area of the shaded region bounded by the curves $y = e^x$, $y = xe^x$, and the vertical line $x = 2$. 2. **Understanding the curves:**

1. The problem is to evaluate the definite integral with given limits. 2. The general formula for a definite integral from $a$ to $b$ of a function $f(x)$ is:

1. **הבעיה:** נתונה המשוואה המעגלית $$25 = y^2 + x^2$$, כאשר $y$ היא פונקציה של $x$, כלומר $y = f(x)$. נדרש למצוא את הנגזרת של $y$ לפי $x$, כלומר $\frac{dy}{dx}$. 2. **הנוסחה והכלל

1. **State the problem:** We need to find the integral of $\sqrt{\tan x}$ with respect to $x$, i.e., $\int \sqrt{\tan x} \, dx$. 2. **Recall the formula and substitution:** This in

1. **State the problem:** We are given two functions $f(x)$ and $g(x)$ from graphs and asked to find various limits involving these functions. 2. **Recall limit rules:** The limit

1. The problem asks for the values of $x_0$ in the interval $-3 \leq x_0 \leq 2$ where the limit $\lim_{x \to x_0} g(x)$ exists. 2. The limit $\lim_{x \to x_0} g(x)$ exists if and

1. **Problem Statement:** Determine for which values of $x_0$ in the interval $-3 \leq x_0 \leq 2$ the limit $\lim_{x \to x_0} g(x)$ exists for the given piecewise linear function

1. The problem is to solve the volume integral, which generally means finding the volume of a solid region using integration. 2. The formula for volume using a triple integral is $

1. **State the problem:** We need to find the length $L$ of the curve defined by the function $$f(y) = 4 \cdot (y + 1)^2$$ over the interval $$3 \leq y \leq 6$$. 2. **Formula for a

1. **State the problem:** We need to find the length $L$ of the curve defined by $y = 2x^{3/2}$ over the interval $0 \leq x \leq 4$. 2. **Formula for arc length:** The length of a

1. **Problem statement:** Find the length $L$ of the curve defined by the function $$f(y) = 3 \cdot (y + 4)^2$$ for $$0 \leq y \leq 4$$. 2. **Formula for arc length:** The length o

1. Masalah yang diberikan adalah mencari hasil integral dari fungsi $\int (3x + 2)^5 \, dx$. 2. Gunakan substitusi untuk memudahkan integral. Misalkan $u = 3x + 2$, maka $du = 3 \,

1. **State the problem:** Find the length of the curve defined by $y = 3 \cdot x^{\frac{3}{2}}$ for $0 \leq x \leq 2$. 2. **Formula for arc length:** The length $L$ of a curve $y=f

1. **State the problem:** Find the derivative of the function $$y = x^2 \sin x$$. 2. **Formula used:** We will use the product rule for derivatives, which states: