∫ calculus

1. **Stating the problem:** We need to find the derivative of the function $$f(x) = \frac{x^3}{x+2}$$ and then multiply it by $$x + \frac{2}{x^3}$$. 2. **Formula used:** To differe

1. **State the problem:** Find the derivative of the function $f(x) = \ln\left(\frac{x^3}{x+2}\right)$.\n\n2. **Recall the formula:** The derivative of $\ln(u)$ with respect to $x$

1. **State the problem:** Find the derivative of the function $f(x) = \ln\left(\frac{x^3}{x+2}\right)$.\n\n2. **Recall the formula:** The derivative of $\ln(u)$ with respect to $x$

1. **State the problem:** Find the derivative of the function $$y=\frac{\ln x^3}{x+2}$$. 2. **Rewrite the function:** Use the logarithm power rule $$\ln x^3 = 3 \ln x$$, so the fun

1. **State the problem:** We want to find the limit $$\lim_{x \to 0} \frac{(1+x)^{\frac{1}{x}} - e}{x}$$ using l'Hopital's rule. 2. **Recall the formula and rules:** The expression

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{(1+x)^{\frac{1}{x}} - e}{x}$$. 2. **Recall the known limit:** We know that $$\lim_{x \to 0} (1+x)^{\frac{1}{x}} = e$

1. **Stating the problem:** We need to sketch and calculate the area bounded by the curves $y=\frac{1}{x}$ and $y=\frac{1}{x^2}$ at $x=2$.

1. **Stating the problem:** We need to find the area bounded by the curves $y = x^2$ and $y = -x$. 2. **Find the points of intersection:** Set $x^2 = -x$ to find where the curves i

1. **Problem Statement:** Find the area bounded by the curves $y=2$, $y=\cos(x)+1$, and the vertical lines $x=0$ and $x=\pi$.

1. **Problem Statement:** Given the equation $$2 \int_0^{10} f(x) \, dx + 8 \int_0^{11} f(x) \, dx + 10 \int_0^{8} f(x) \, dx = 9,$$ find the value of $$2 \int_0^{11} f(x) \, dx.$$

1. **State the problem:** We are given the rate of change of the area of a lamina as $$\frac{dA}{dt} = e^{-0.2t}$$ where $t$ is in seconds. The initial area at $t=0$ is 140 cm². We

1. **Problem Statement:** Given a curve $y=f(x)$ with second derivative $$\frac{d^2y}{dx^2} = ax + b,$$ where $a$ and $b$ are constants. The curve has an inflection point at $(0,2)

1. **Stating the problem:** We want to find the intervals where the function $f(x) = \tan x - x$ is increasing (croissante) and decreasing (décroissante). 2. **Formula and rules:**

1. **Problem Statement:** Find the function $y = f(x)$ whose slope of the tangent at any point $(x,y)$ is given by the derivative $\frac{dy}{dx} = -6 \cos^5 x \sin x$, and the curv

1. **Problem Statement:** We are given the slope of the tangent to the curve $f$ at any point $(x,y)$ as $g(x) = \frac{xe^x}{(x+1)^2}$.

1. **State the problem:** We are given a function defined by an integral: $$f(x) = \int (x + 1)(2x^2 + 4x - 1) \, dx$$ with the initial condition $$f(-2) = 1$$. We need to find the

1. The problem asks us to identify which graph represents the curve of the second derivative $f''(x)$ given the graph of the first derivative $f'(x)$. 2. Recall that $f''(x)$ is th

1. **State the problem:** Given the differential equation $$\frac{dy}{dx} = \csc^2 x$$ with the initial condition $$y\left(\frac{\pi}{4}\right) = 2$$, find the value of $$y$$ when

1. **Problem statement:** Calculate the definite integral $$\int_0^{20 \pi} |\sin x| \, dx$$. 2. **Formula and important rules:** The integral of the absolute value of sine over on

1. **Stating the problem:** We are given the integral $$\pi \int_{-2}^2 (4 - x^2) \, dx$$ and asked to identify which solid's volume it represents. 2. **Understanding the integral:

1. **Problem 2:** Find the equation of the tangent line to the graph of a function at the point $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$. To find the tangent line, we