∫ calculus

1. **Énoncé du problème :** Déterminer l’ensemble des fonctions primitives de la fonction $f_1(x) = 2x(x^2 - 3)^5$. 2. **Formule utilisée :** Pour trouver une primitive, on utilise

1. **State the problem:** We want to find the limit $$\lim_{n \to \infty} \frac{\tan^2 n}{n}.$$\n\n2. **Recall the behavior of the functions:** The numerator is $\tan^2 n$, which o

1. **Problem (三):** Find values of $a$ and $b$ such that $(2,3)$ is an inflection point of $y = ax^3 + bx^2$. 2. **Recall:** An inflection point occurs where the second derivative

1. **問題陳述**： (五) 求極限 $$\lim_{n \to \infty} \frac{\sqrt{1} + \sqrt{2} + \cdots + \sqrt{n}}{\sqrt{n^3}}$$，利用黎曼積分方法。

1. **State the problem:** We want to find the limit $$\lim_{x \to 0} \left( \frac{1}{\sin^2 x} - \frac{1}{x^2} \right).$$ 2. **Recall important formulas and rules:**

1. **Problem Statement:** We are given the function $f(x) = 1 + \frac{1}{x} + \frac{1}{x^2}$ and want to analyze its vertical and horizontal asymptotes, critical points, and inflec

1. **Problem Statement:** We have a piecewise function defined on the interval $[-2,2]$:

1. **Problem:** Find the value of $a$ such that $$\lim_{x \to +\infty} \left(\frac{x+a+1}{x+1}\right)^{x-1} = \frac{1}{\sqrt{e}}.$$ 2. **Formula and rules:** Recall the limit defin

1. **Find the arc length of** $y = \frac{x^4}{8} + \frac{1}{4x^2}$ from $x=1$ to $x=2$. 2. **Find the arc length of** $(y-1)^3 = x^2$ on the interval $[0,8]$.

1. **State the problem:** Water is flowing out of a conical tank at a rate of 4 cm³/s. The tank has radius 4 cm and height 10 cm. We want to find how fast the radius of the water s

1. Problem: Find the indefinite integral \(\int (x^3 + \sqrt{x}) \, dx\). 2. Formula: The integral of a sum is the sum of the integrals, and \(\int x^n \, dx = \frac{x^{n+1}}{n+1}

1. **Problem statement:** Given the drug entry rate function $$R(t) = \frac{80t}{t^2 + 4}$$ for $$t \geq 0$$, answer the following: (a) Find the rate of drug entering the bloodstre

1. **Problem:** Find the equations of the tangent and normal lines to the curve $$y = 3x^4 - 5x^3 + 6x + 8$$ at the point where $$x=1$$. 2. **Formula and rules:**

1. **State the problem:** Find the limit of the function as the variable approaches a certain value. 2. **Formula and rules:** The limit of a function $f(x)$ as $x$ approaches $a$

1. **Problem statement:** We have a region bounded by the curves $y=4x - x^2$ and $y=x$, rotated about the vertical line $x=4$. We need to find:

1. **Problem statement:** We have the region bounded by the curves $y=4x - x^2$ and $y=x$, rotated about the line $x=4$. We need to find:

1. **Problem:** Evaluate the iterated integral $$\int_0^1 \int_0^2 (x + 3) \, dy \, dx$$ 2. **Formula and rules:** For iterated integrals, integrate the inner integral first with r

1. **Problem:** Evaluate the limit $$\lim_{t \to \infty} \frac{\ln(3t)}{t^2}$$ using L'Hospital's Rule. 2. **Formula and rule:** L'Hospital's Rule states that if $$\lim_{x \to a} \

1. **State the problem:** Find the derivative $f'(x)$ of the function $f(x) = \frac{1}{-3x - 3}$ using the definition of the derivative. 2. **Recall the definition of the derivativ

1. The problem is to find the partial derivative $\frac{\partial}{\partial x}$ of the function $y(1+y^2)$ where $y$ is implicitly defined by the equation $1,0 = \sqrt{x}$ (interpre

1. The problem is to evaluate the integral $$\int_0^{\frac{\pi}{2}} \cos^3 x \, dx$$. 2. We use the reduction formula or rewrite the power of cosine using trigonometric identities.