∫ calculus

1. **Problem statement:** Find the derivative of the function $f(x) = 2x^2 - x + 5$ and then evaluate it at $x=3$. 2. **Formula and rules:** The derivative of a function $f(x)$, de

1. **Problem statement:** We need to find the area $A$ of the gray shaded region between the graphs of the quadratic function $f$ and the linear function $g$ from $x=0$ to $x=6$. 2

1. Het probleem: Je vraagt waarom er een $x$ verschijnt bij de term $4e^2$ in de primitieve stap van de integraal. 2. Formule en uitleg: Bij het integreren van een constante functi

1. **Probleemstelling:** Je wilt begrijpen wat er bedoeld wordt met de 'buitenste' en 'binnenste' inhoud bij het berekenen van het volume van een lichaam dat ontstaat door het went

1. Het probleem is om te begrijpen hoe de haakjes zijn weggewerkt van de eerste naar de tweede stap in de integraal: $$\pi \int_0^p \left(e^{\frac{1}{2}x + 1}\right)^2 dx = 9\pi e^

1. **State the problem:** Find the derivative of the function $f(x) = |x + 7|$. 2. **Recall the formula:** The absolute value function $|u|$ can be written as $|u| = \sqrt{u^2}$, b

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