∫ calculus

1. **Problem:** Find the critical points of $F(x) = x^3 - 3x^2$ and classify them as maxima or minima. 2. **Step 1: Find the first derivative**

1. **State the problem:** Find the limit as $x$ approaches $+\infty$ of the function $\ln\left(\frac{x+1}{x}\right)$. 2. **Recall the limit and logarithm properties:** The natural

1. Stating the problem: Evaluate the definite integral $$\int_{\frac{\pi}{2}}^{0} \frac{1+\cos 2t}{2} \, dt$$. 2. Formula and rules: Recall the integral of cosine and the property

1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{(5 - 2x)^2 - 243}{x - 1}$$. 2. **Recall the formula and approach:** This is a limit of the form $$\frac{f(x) - f(a)}

1. We are asked to find the limit as $n$ approaches 0 of the expression $$\frac{1-\cos 2n}{n}$$. 2. Recall the trigonometric identity: $$1-\cos x = 2\sin^2 \frac{x}{2}$$.

1. 题目要求计算积分 $$\int x \arctan x \, dx$$。 2. 使用分部积分法，设 $$u = \arctan x$$，则 $$du = \frac{1}{1+x^2} dx$$。

1. **State the problem:** We need to evaluate the integral $$\int t \cos x \, dx$$. 2. **Identify variables:** Here, $t$ is treated as a constant with respect to $x$ because the in

1. **State the problem:** Find the limit as $x \to \infty$ of $$\frac{(3^x - 2)^2}{2 \cdot 9^x}$$. 2. **Rewrite the expression:** Note that $9^x = (3^2)^x = 3^{2x}$. So the express

1. **State the problem:** Find the limit \( \lim_{x \to -\infty} 4(5^x) + 2 \). 2. **Recall the behavior of exponential functions:** For any base \( a > 1 \), \( a^x \to 0 \) as \(

1. **State the problem:** We want to find the limit as $x$ approaches 0 of the expression $\left(1+4\sin x\right)^{\cot x}$. 2. **Recall the limit form:** This is a limit of the fo

1. Differentiate the function $y = 3x^2 - x^4 + 2$. Step 1: State the problem.

1. **State the problem:** Find the limit $$\lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9}$$. 2. **Recognize the indeterminate form:** Substituting $x=9$ directly gives $$\frac{\sqrt{9}

1. **State the problem:** Find the limit as $x \to +\infty$ of the rational function $$\frac{3x^5 - 5x + 6}{-6x^5 - 2x^4 + x + 3}.$$\n\n2. **Recall the rule for limits of rational

1. The problem involves understanding the expression for the integrand given as $2x = 2 \times 1 = 2$ and then interpreting $2x = 2t$. 2. First, the integrand is the function insid

1. **State the problem:** We need to evaluate the line integral $$\int_C 2x \, ds$$ where the curve $C$ consists of two parts: - $C_1$: the parabola $y = x^2$ from $(0,0)$ to $(1,1

1. **State the problem:** Evaluate the definite integral $$\int_0^{\frac{\pi}{4}} \tan x \sec^2 x \, dx$$ using substitution. 2. **Recall the substitution method:** We look for a s

1. **State the problem:** Find the limit $$\lim_{x \to \pi} \frac{\sin x}{2 + \cos x}$$. 2. **Recall the limit rule:** If the function is continuous at the point, the limit is the

1. **State the problem:** We need to explain why the function \(f(x) = \frac{1}{x+2}\) is discontinuous at \(a = -2\). 2. **Recall the definition of continuity at a point:** A func

1. **State the problem:** Given the function $f(n,y) = n^2 \arctan^{-1}\left(\frac{y}{n}\right)$, find the second order partial derivatives $\frac{\partial^2 f}{\partial n^2}$ and

1. The problem asks for the limit of the function $f(x)$ as $x$ approaches 6. 2. The limit of $f(x)$ as $x$ approaches a value $a$ is the value that $f(x)$ gets closer to as $x$ ge

1. Problem 25: Evaluate $$\int_0^{\pi/4} (1 + e^{\tan \theta}) \sec^2 \theta \, d\theta$$ using substitution. 2. Use substitution: let $$u = \tan \theta$$, then $$\frac{du}{d\theta