∫ calculus

1. The problem asks if the function defined on the interval $(1,2]$ has a maximum value. 2. To determine this, we need to know the function's behavior on the interval $(1,2]$.

1. **State the problem:** Prove that the curve defined parametrically by $$x=a[\cos(\theta) + \theta \sin(\theta)], \quad y=a[\sin(\theta) - \theta \cos(\theta)]\n$$ for $a > 0$ an

1. We are asked to find the antiderivative of \(4x \cos(2 - 3x)\) using integration by parts if appropriate.\n\n2. Let \(u = 4x\) and \(dv = \cos(2 - 3x) dx\). Then \(du = 4 dx\) a

1. The problem is to evaluate the double integral $$\int_3^5 \int_1^2 x^2 y \, dy \, dx$$. 2. First, integrate the inner integral with respect to $y$, keeping $x$ fixed:

1. The problem is to find the derivative of the function $$f(x)=\frac{1-\cos(0x)}{1-(1+\tanh^5(x))}$$ with respect to $x$. 2. Simplify the function first:

1. مسئله: حد $$\lim_{x \to 0} \frac{x - \sin(x)}{x - \tan(x)}$$ را با استفاده از قاعده هوپیتال پیدا کنید. 2. ابتدا بررسی می‌کنیم که صورت و مخرج در نقطه $x=0$ به چه مقداری میل می‌کن

1. The problem asks us to find the indefinite integral of the function $x^2$, which means finding a function $F(x)$ whose derivative is $x^2$. 2. Recall the power rule for integrat

1. Trouver $\min_{x \in \mathbb{R}} \left[(x^2 - 1)^2 + 3\right]$ et $\operatorname{Argmin} \{(x^2 - 1)^2 + 3; x \in \mathbb{R}\}$. - L'expression est $f(x) = (x^2 - 1)^2 + 3$.

1. Let's understand the problem: You want to know how to use the *same method* in integration for different integrals. 2. The key idea is to identify the method you want to reuse,

1. The problem is to find the derivative of the function $$f(x) = x^{4^{3x}}.$$\n\n2. Recognize that the given function is an exponential function where the exponent itself is an e

1. **Problem:** Find the derivative of the function $$f(x) = x^{4 - x^2}$$ using logarithmic differentiation. 2. **Step 1: Apply logarithm to both sides**

1. Given the function $f(x)=x^{\sin x}$, we want to find its derivative $f'(x)$ using logarithmic differentiation. 2. Start by taking the natural logarithm of both sides:

1. **State the problem:** Find the equation of the tangent line to the curve $f(x) = 3^x$ at $x=1$. 2. **Find the point of tangency:** Evaluate $f(1)$ to get the $y$-coordinate.

1. Calculer les primitives des fonctions suivantes : 1. $f(x)=\sqrt{1+x^2}$

1. Problem 16: Solve the first-order vector differential equation $$\frac{d\mathbf{r}}{dt} = \frac{t}{t^2+2}\mathbf{i} - \frac{t^2+1}{t-2}\mathbf{j} + \frac{t^2+4}{t^2+3}\mathbf{k}

1. **State the problem:** We want to find values of $a$ and $b$ such that the piecewise function $$f(x) = \begin{cases} x^2 + a, & x \geq 3 \\ bx + a, & -3 < x < 3 \\ \sqrt{-b - x}

1. We are asked to find \(\lim_{x \to -1} f(x)\). Looking at the graph, the function on the interval including \(x=-1\) is a straight line segment from (-3, -1) to (-1, 3). 2. To f

1. **State the problem:** Find the limit of $f(x)$ as $x$ approaches 1, i.e., $\lim_{x \to 1} f(x)$, and then check continuity at $x = 1$ by comparing the left-hand limit, right-ha

1. **Problem:** Air is pumped into a spherical balloon at 5 cm³/min. Find the rate of change of the radius when diameter is 20 cm. Step 1: Volume of sphere is $$V=\frac{4}{3}\pi r^

1. **State the problem:** We need to find the limits of the function $f(x)$ approaching $x=-1$ from the left, from the right, and overall at $x=-1$. 2. **Find the left-hand limit $

1. We are asked to evaluate the limit \(\lim_{x \to 1} \left( \frac{1}{1-x} - \frac{3}{1-x^3} \right)\).\n\n2. First, notice that as \(x \to 1\), the denominators \(1-x\) and \(1-x