∫ calculus

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

1. **State the problem:** Find the limit as $x$ approaches 3 of the expression $$\frac{x^2 - 9}{x - 3}$$. 2. **Recall the formula and rules:** When direct substitution leads to an

1. The problem is to find the indefinite integral of the function $5$ with respect to $x$, i.e., $\int 5 \, dx$. 2. The formula for integrating a constant $a$ with respect to $x$ i

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

1. **State the problem:** Find the first derivative of the function $$y(x) = k e^{x^3} + f$$ with respect to $$x$$. 2. **Recall the chain rule:** If $$y = g(h(x))$$, then $$\frac{d

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. **State the problem:** Find the first derivative of the function $$y(x) = k e^{x^3} + f$$ with respect to $$x$$. 2. **Recall the chain rule:** If $$y = g(h(x))$$, then $$\frac{d

1. **Problem:** Differentiate the function \(g(x) = (x^2 - 2)(2x + 3)\). 2. **Formula:** Use the product rule for differentiation: \(\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)\

1. **Stating the problem:** Differentiate the function $$f(x) = \frac{3x^2 - x}{\sqrt{1-2x}}$$ using the quotient rule. 2. **Recall the quotient rule:** For $$f(x) = \frac{u(x)}{v(

1. **State the problem:** Find the area under the curve $y = 2\sqrt{x}$ from $x=2$ to $x=7$. 2. **Formula used:** The area under a curve $y=f(x)$ from $x=a$ to $x=b$ is given by th

1. **Problem statement:** We analyze the water volume function $f(t)$ over time $t$ hours, given a graph with volume in cubic meters.

1. **State the problem:** Evaluate the integral $$\int_1^{600} (15^x - 12^x) \, dx + \int_2^{600} (12^x - 15^x) \, dx.$$\n\n2. **Rewrite the expression:** Combine the integrals as\

1. **State the problem:** We are given two definite integrals: $$\int_{-1}^4 f(x) \, dx = -2$$

1. **State the problem:** Find the limit $$\lim_{h \to 0} \frac{5e^x - 5e^{x+h}}{3h}$$ without using a calculator.

1. The problem asks for the instantaneous rate of change of the function \( g(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h} \) at \( x = \frac{\pi}{3} \).\n\n2. Recognize that \

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 $$\int f(g(x)) g'(x) \, dx$$.