∫ calculus

1. The problem is to find the derivative operator $\frac{d}{dx}$, which represents the process of differentiating a function with respect to $x$. 2. The derivative of a function $f

1. **Énoncé du problème** : Trouver une primitive $G$ de la fonction $g$ définie sur $\mathbb{R}_+$ par $$g(x) = \frac{e^{\frac{1}{x}}}{x^2}$$ telle que $G(1) = 0$. 2. **Formule et

1. **State the problem:** Evaluate the integral $$\int_0^1 8 \, dx$$ and then add $$\sqrt{8}$$ to the result. 2. **Recall the formula for definite integrals:**

1. **State the problem:** Evaluate the definite integral $$\int_2^5 \frac{x}{\sqrt{x-1}} \, dx$$. 2. **Rewrite the integral:** Let us express the integral in a simpler form:

1. **Stating the problem:** We need to find the result of the operation $+\infty - x$ where $x \in \mathbb{R}$. 2. **Understanding the operation:** $+\infty$ represents an infinite

1. Il problema chiede di trovare i limiti della funzione $f(x)$ in vari punti e verso l'infinito, basandosi sul grafico descritto. 2. Ricordiamo che il limite di una funzione in un

1. **State the problem:** We need to find the integral $$\int x^5 \ln(x^2 + 2x + 4) \, dx$$. 2. **Formula and approach:** Use integration by parts, where $$\int u \, dv = uv - \int

1. **State the problem:** We need to solve the integral $$\int \frac{1}{x \sqrt{x^4 - 4}} \, dx.$$\n\n2. **Identify substitution and formula:** Notice the expression under the squa

1. **Problem:** Find the length of the curve given by $$\mathbf{r}(t) = \sqrt{2}t\mathbf{i} + e^t\mathbf{j} + e^{-t}\mathbf{k}$$ for $$0 \leq t \leq 8$$. 2. **Formula for curve len

1. **State the problem:** We need to find the value of $x$ where the piecewise function $$f(x) = \begin{cases} 5x, & x < 0 \\ 1, & x = 0 \\ -5x, & x > 0 \end{cases}$$

1. To determine if a function is increasing or decreasing, we first need to know the function itself. 2. If the function is given as $y=f(x)$, we find its derivative $f'(x)$.

1. **State the problem:** Find the area of the region $R$ inside both polar curves $r=4\cos\theta$ and $r=5-4\cos\theta$ in the first quadrant. 2. **Find the points of intersection

1. **State the problem:** Evaluate the integral $$\int_0^{\frac{\pi}{6}} (1 - \cos 3x) \sin 3x \, dx$$. 2. **Recall the formula and rules:** We will use substitution and trigonomet

1. **State the problem:** Calculate the area of the region bounded by the curves $g(x) = \sqrt{x - 1}$ and $k(x) = x - 3$ from $x=1$ to $x=5$.

1. **Problem statement:** Find the integral expression for the volume of the solid formed by rotating the region bounded by the curve $f(x) = 2x - 1$, the x-axis, and the vertical

1. **State the problem:** We want to find the differential formulas that estimate the change in volume $dV$ of a sphere when its radius changes by a small amount $dr$ from an initi

1. **Problem statement:** We are given that the function $f$ passes through the point $(5,3)$ and that its derivative at $x=5$ is $f'(5)=3$. We want to find the equation of the tan

1. **Problem statement:** Given that $f'(3) = 5$ and $f(3) = 5$, find the equation of the tangent line to the graph of $f$ at the point $P(3, f(3))$. 2. **Formula used:** The equat

1. **State the problem:** Differentiate the expression $$b + wL + F - c - q(y - x) - \frac{Rb}{a} - \frac{pq}{2} \left(\frac{y - x}{x}\right)^2 x$$

1. **Problem Statement:** Evaluate the definite integral $$\int_3^6 g(x) \, dx$$ where the graph of $$g$$ from $$x=2$$ to $$x=6$$ is a line segment rising linearly from approximate

1. **Problem Statement:** Evaluate the definite integrals of the piecewise linear function $f(x)$ using the geometry of the graph.