∫ calculus

1. **State the problem:** Find the limits of the function $$f(x) = \sqrt{x^2 + 5x - x}$$ as $$x \to \infty$$ and $$x \to -\infty$$. 2. **Simplify the function inside the square roo

1. **State the problem:** Find the limits of the function $$f(x) = \frac{\sqrt{x+5} - 3}{x - 4}$$ as $$x \to \infty$$ and $$x \to -\infty$$. 2. **Recall the limit rules:** When $$x

1. **Problem statement:** Given a function $f$ with continuous second derivative on $\mathbb{R}$, and conditions $f'(0)=-1$, $f(1)=0$, and $f'(1)=0$, prove that $$\int_0^1 f(x)\cos

1. **State the problem:** Find the limit $$\lim_{x \to 0} x \frac{f(5+x) - f(5)}{x}$$ where $$f(x) = 2x^2 + 3$$. 2. **Rewrite the expression:** The limit can be simplified as $$\li

1. **Problem statement:** We are given two functions $f(x)$ and $g(x)$ with their graphs and asked to evaluate various limits involving these functions. 2. **Recall limit propertie

1. **State the problem:** Find the limit as $x$ approaches 8 of the function $f(x) = (3 + \sqrt[3]{x})(2 - 5x^2 + x^3)$.

1. The problem asks to solve all given problems using integrals, but since no specific problems are provided, I will demonstrate how to solve a basic integral problem. 2. Consider

1. **Probleemstelling:** We bekijken de grafiek van de afgeleide functie $f'(x)$ en moeten de mogelijke buigpunten, extrema en het verloop van de originele functie $f(x)$ bepalen.

1. **State the problem:** We need to find the limit of the function $$f(x) = 7 - 2e^{-3x}$$ as $$x$$ approaches infinity. 2. **Recall the limit rule for exponential decay:** When $

1. **Stating the problem:** Evaluate the integral $$\int \frac{3x - \sqrt{Bgsin x}}{\sqrt{1 - x^2}} \, dx$$.

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the curve $y = 4x \sqrt{3x - 2}$ for $x \geq \frac{2}{3}$, and then find the equation of the tangent line at the po

1. **State the problem:** Differentiate the function $f(x) = \tan x$ with respect to $x$. 2. **Recall the formula:** The derivative of $\tan x$ is given by the formula

1. **State the problem:** We need to evaluate the integral $$\int_{-2\pi}^{2\pi} x \cos x \, dx$$. 2. **Recall the formula and rules:** To solve this integral, we use integration b

1. **State the problem:** Evaluate the definite integral $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} x \sin x \, dx$$. 2. **Recall the formula and properties:** The integral of a produ

1. **State the problem:** We need to find the integral $\int x \sin x \, dx$. 2. **Formula and method:** Use integration by parts, which states:

1. **State the problem:** We are given the function $$h(x) = x^3 - 6x^2 + 7x + 6$$ and asked to find:

1. **State the problem:** We need to evaluate the integral $$\int 12 \cos^2 x \sin x \, dx$$. 2. **Recall the formula and substitution:** Notice the integral involves powers of cos

1. **Problemstellung:** Berechne das Integral $$\int_{-1}^2 -2t \, dt$$ mithilfe von Dreiecks- und Rechtecksflächen. 2. **Formel und Erklärung:** Das Integral einer linearen Funkti

1. **State the problem:** Calculate the definite integral $$\int_3^9 x \, dx$$. 2. **Recall the formula:** The integral of $$x$$ with respect to $$x$$ is $$\frac{x^2}{2} + C$$, whe

1. **State the problem:** We want to evaluate the double integral $$\iint_R \sqrt{x^2 + y^2} \, dA$$ where the region $$R$$ is the disk defined by $$x^2 + y^2 \leq 4$$.

1. **بیان مسئله:** ما تابع $$f(x,y) = \ln\left(\frac{\sqrt{4 - x^2 - y^2}}{|x - y|}\right) + \arcsin(xy)$$ را داریم و می‌خواهیم: