∫ calculus

1. **Problem:** Determine which of the functions $F_1(x) = \cos x$, $F_2(x) = -\cos x$, $F_3(x) = \sin x$, $F_4(x) = -\sin x$ is an antiderivative of $f(x) = -\cos x$ on $x \in (-\

1. مسئله را بیان می‌کنیم: تابع $f(x)$ به صورت $$f(x) = \int_2^x \frac{\cos(\pi t)}{1+t^2} dt$$

1. **State the problem:** Find the volume of the solid generated by rotating the area in the first quadrant of the ellipse $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$ about the x-axis th

1. **State the problem:** Find the volume of the solid generated when the area in the first quadrant of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is revolved around the x

1. **State the problem:** Evaluate the definite integral $$\int_1^4 x^2 \ln(x) \, dx$$. 2. **Recall the integration by parts formula:**

1. The problem is to show that $$\int a^x \, dx = \frac{a^x}{\ln a} + C$$ where $a > 0$ and $a \neq 1$. 2. Recall the formula for the integral of an exponential function with base

1. Problem: Izračunaj integral funkcije $f(x)$, ki ni podana, zato predpostavimo splošni primer integrala $\int f(x) \, dx$. 2. Formula: Za izračun določenega ali nedoločenega inte

1. Het probleem is om de afgeleide te vinden van de functie $$y = 5 \cdot 4^{-x^2 + 5x - 2}$$. 2. We gebruiken de kettingregel en de afgeleide van een exponentiële functie met basi

1. **State the problem:** We need to find the limit $$\lim_{x \to +\infty} \frac{\ln(x^2 + 3x + 2) - \ln(x^2 - 4x + 2)}{x^2}.$$\n\n2. **Use logarithm properties:** Recall that \(\l

1. **State the problem:** Find the limit $$\lim_{x \to \infty} \frac{\ln(x^2 + 3x + 2) - \ln(x^2 - 4x + 2)}{x^2}$$

1. **Stating the problem:** We analyze the limits and vertical asymptotes of the function $A(x)$ given the behavior near $x = -3$, $x = 2$, and $x = -1$. 2. **Limits at vertical as

1. The problem asks for the equations of the vertical asymptotes of the given graph. 2. Vertical asymptotes occur where the function approaches infinity or negative infinity as $x$

1. **Stating the problem:** Evaluate the double integral

1. **State the problem:** We are given the function $f(x) = x^3 - 15x$ and need to determine whether it has maximum or minimum values and understand its impact on network efficienc

1. El problema es calcular la integral $$I = \int x (a - bx^2) \, dx$$ donde $a$ y $b$ son constantes. 2. Usamos la propiedad distributiva para expandir el integrando:

1. **Problem:** Determine whether the series $$\sum_{n=1}^\infty \frac{2 - \sin^2(n)}{e^n}$$ converges or diverges. 2. **Test and formula:** We use the Comparison Test. Since $\sin

1. **State the problem:** We are given the function $f(x) = x^3 - 6x^2 + 9x + 2$ and asked to find intervals where $f$ is increasing or decreasing using the graph and then verify b

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{e^{2x} - 1}{x}$$. 2. **Recall the formula and rules:** The limit $$\lim_{x \to 0} \frac{e^{kx} - 1}{x} = k$$ for any

1. **Problem statement:** Given velocity function $v(t) = - (t + 1) \sin\left(\frac{t^2}{2}\right)$ and initial position $x(0) = 1$, find:

1. **Problem statement:** A particle moves along the x-axis with velocity given by

1. **State the problem:** We have a particle moving along a horizontal line with position function $$s(t) = t^3 - 9t^2 + 15t + 4$$ for $$t \geq 0$$. We want to find when the partic