∫ calculus

1. **State the problem:** Differentiate implicitly the function $$y=\frac{\ln((2x+4)^9(4x-2))}{\ln\sqrt[5]{3x-1}}$$ to find $$\frac{dy}{dx}$$. 2. **Rewrite the function for clarity

1. **Problem statement:** Differentiate the following functions: a) $g(x) = (x^2 - 2)(2x + 3)$

1. Énonçons le problème : Calculer l'intégrale $$\int \frac{2x^3}{\sqrt{4-x^2}} \, dx$$. 2. Pour résoudre cette intégrale, on peut utiliser une substitution trigonométrique car l'e

1. **Problem statement:** A drone's velocity is given by $$v(t) = 4e^{-0.3t} \sin(1.2t)$$ for $$0 \leq t \leq 12$$ seconds. We need to find: (a) The maximum speed of the drone.

1. **State the problem:** We are given the function $$f(x) = 6 + x^2 + \sin 4x$$ and need to find its derivative $$f'(x)$$ with respect to $$x$$. 2. **Recall the derivative rules:*

1. Problema: Calcular a integral $$\int x \tan(x^2) \, dx$$. 2. Fórmula e regra: Usamos substituição para integrais do tipo $$\int f(g(x)) g'(x) \, dx$$.

1. **Problem Statement:** We are given a function $y=f(x)$ with a graph and asked to find: a) The smallest positive number $h$ such that $\frac{f(1+h)-f(1)}{h} = 0$.

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$. 2. **Recall the formula and rules:** When direct substitution results in an indeterminate form li

1. **بيان المسألة:** لدينا الدالة $$F(x) = \frac{2x + 2}{x - 1}$$

1. **State the problem:** We need to find the difference between the upper and lower sums for the function $f(x) = 3^x$ over the interval $[0,2]$ using four subintervals. 2. **Form

1. **State the problem:** Calculate the surface area of the solid formed by revolving the curve $f(x) = \frac{1}{5}x^3$ around the x-axis from $x=3$ to $x=5$. 2. **Formula for surf

1. The problem asks to find the value of $x$ for which the volume of the cuboid is maximum. 2. The volume function $V(x)$ is given or derived as a quadratic expression. To find the

1. **State the problem:** We want to find the intervals where the function $$F(x) = \int_x^0 V'(t) P(t) \, dt$$ is strictly increasing. 2. **Recall the Fundamental Theorem of Calcu

1. **State the problem:** Evaluate the triple integral $$\int_{-1}^2 \int_1^{y+2} \int_e^{2e} \frac{e^{2e^{x+y}}}{z} \, dz \, dx \, dy.$$\n\n2. **Inner integral:** Integrate with r

1. **State the problem:** Evaluate the triple integral $$\int_{-1}^2 \int_1^{y+2} \int_0^{e^{2e^{x+yz}}} dz\,dx\,dy$$ (interpreting the user's integral as $$\int_{-1}^2 \int_1^{y+2

1. **State the problem:** Find the limit of the function $f(x) = e^{\frac{1}{x+2}}$ as $x$ approaches 2. 2. **Recall the limit definition and properties:** The exponential function

1. **State the problem:** We have a drug amount function $$C(t) = -t^3 + 4.5t^2 + 54t$$ representing the amount of drug in the bloodstream after $$t$$ hours.

1. **State the problem:** Evaluate the triple integral $$\int_{-1}^2 \int_1^{y+2} \int_e^{2e} \frac{(x+y)}{z} \, dz \, dx \, dy.$$ 2. **Inner integral:** Integrate with respect to

1. **State the problem:** Evaluate the triple integral $$\int_{-1}^2 \int_{1}^{y+2} \int e^{2ex+yz} \, dz \, dx \, dy.$$ 2. **Inner integral:** Given $$\int e^{2ex+yz} \, dz,$$ tre

1. The problem is to find the primitive (antiderivative) of the function $$f(x) = \frac{\cos^2(2x)}{5 - x \sin(x)}$$. 2. The primitive or antiderivative of a function $f(x)$ is a f

1. **Problem statement:** Find the intercepts, asymptotes, critical points, intervals of increase/decrease, inflection points, concavity, and relative extrema for the function $$y