∫ calculus

1. **Énoncé du problème :** Calculer le volume du solide de révolution obtenu en faisant tourner la région bornée par $y = x^2$, $y = 4$, et $x = 0$ (avec $x \geq 0$) autour de dif

1. **State the problem:** We want to find the average value of the function $$f(t) = 72 - 60\cos\left(\frac{\pi}{3}t\right)$$ over the interval $$[0,8]$$. 2. **Formula for average

1. **State the problem:** Differentiate the function $$y = (\ln(3x))^{17}$$ with respect to $$x$$. 2. **Formula and rules:** We will use the chain rule for differentiation, which s

1. The problem is to understand how the mathematical constant $e$ is derived. 2. The number $e$ is defined as the limit of \(\left(1 + \frac{1}{n}\right)^n\) as $n$ approaches infi

1. The problem asks us to determine for which function the value of $y$ approaches $-\infty$ as $x$ approaches $\infty$. 2. Let's analyze each function one by one:

1. **State the problem:** Evaluate the integral $$\int x^5 \cos(x^3) \, dx$$ using substitution and integration by parts. 2. **Substitution:** Let $$u = x^3$$. Then, $$du = 3x^2 \,

1. **State the problem:** Calculate the definite integral $$\int_0^3 x e^{-x} \, dx$$. 2. **Formula and method:** Use integration by parts, where $$\int u \, dv = uv - \int v \, du

1. **State the problem:** Calculate the definite integral $$\int_0^1 \frac{-2x}{e^{2x}} \, dx$$. 2. **Rewrite the integral:** Note that $$\frac{1}{e^{2x}} = e^{-2x}$$, so the integ

1. **State the problem:** We want to evaluate the definite integral $$\int_0^2 (12x + 4) f'(x) \, dx$$ given that the area bounded by the graph of $$y = f(x)$$, the x-axis, and the

1. **State the problem:** Evaluate the integral $$\int x^8 \ln(x) \, dx$$ using integration by parts. 2. **Recall the integration by parts formula:**

1. **State the problem:** Find the indefinite integral of the function $3x \cos(3x)$. 2. **Formula and method:** We will use integration by parts, which states:

1. **State the problem:** We want to solve for the integral $$\int e^{4x} \sin(5x) \, dx$$ given the system: $$\int e^{4x} \sin(5x) \, dx = -\frac{1}{5} e^{4x} \cos(5x) + \frac{4}{

1. **Problem statement:** Find the derivative of the function \(y = (1 + 3x + 4x^2)^{-3}\). 2. **Formula and rules:** Use the Chain Rule for derivatives of composite functions. If

1. The problem is to understand how the factor $\frac{1}{9}$ is taken out in the expression $$\frac{1}{3} x e^{3x} - \frac{1}{9} e^{3x} + C = \frac{1}{9} (3x - 1) e^{3x} + C.$$\n\n

1. **State the problem:** Find the volume of the solid obtained by rotating the region bounded by $y=7x^2$, $x=2$, $x=3$, and $y=0$ about the x-axis. 2. **Formula used:** The volum

1. **Problem Statement:** Find a reduction formula for the integral $$\int \sin^n x \, dx$$ where $n$ is a positive integer. 2. **Formula and Rules:** We use integration by parts a

1. **Problem statement:** We have a rectangle ABCD with vertices A and B on the x-axis at points (-x,0) and (x,0), and vertices C and D on the curve $y=3e^{-x^2}$. We want to find

1. **State the problem:** We are given the derivative of the plant height function as $$P'(x) = 1 \cdot 1 + 2 \cdot 73 x - 0 \cdot 078 x^2$$ and need to find the range of values of

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{x^2 - x - 2}{x^2 - 6x + 9}$$. 2. **Recall the formula:** For a function $$f(x) = \frac{u(x)}{v(x)}$$, t

1. **Enunciado do problema:** Considere as funções

1. Problemet handlar om att bestämma volymen av en rotationskropp som bildas när området begränsat av kurvan $y=\ln x$, linjen $y=2$ och de positiva koordinataxlarna roterar kring