∫ calculus

1. **Problem:** Calculate the definite integral $$\int_{-2}^1 \frac{4}{x^2} \, dx$$. 2. **Formula and rules:** The integral of $$\frac{1}{x^2}$$ is $$\int x^{-2} dx = -x^{-1} + C =

1. **State the problem:** We need to evaluate the integral $$\int x e^{-2x} \, dx$$. 2. **Formula and method:** We will use integration by parts, which states:

1. **State the problem:** Find the limit as $x$ approaches $+\infty$ of the expression $$\lim_{x \to +\infty} \frac{3 - x}{\sqrt[3]{8x^3 - 3x^2 + x - 4}}.$$\n\n2. **Recall the form

1. **Problem statement:** Evaluate the integral $$\int_{-1}^1 f(x) \, dx$$ where $$f(x) = e^{\sqrt[3]{x}} - 4x$$, given that $$\int_0^1 f(x) \, dx = 3e - 8$$ and $$\int_0^{-1} f(x)

1. The problem is to evaluate the definite integral $$\int_1^7 x \, dx$$. 2. The formula for the integral of $$x$$ is $$\int x \, dx = \frac{x^2}{2} + C$$, where $$C$$ is the const

1. **State the problem:** Differentiate the function $$y = 7x^5 - 4x^3 + 9x - 2$$ with respect to $$x$$. 2. **Recall the power rule for differentiation:** If $$y = x^n$$, then $$\f

1. **State the problem:** Given the temperature function during illness $$T(t) = -0.6t^2 + 0.67t + 37$$ where $T$ is temperature in degrees Celsius and $t$ is time in days, we need

1. **State the problem:** Find the integral of $5x\cos x\,dx$. 2. **Formula and method:** Use integration by parts, which states:

1. The problem is to create a cheat sheet for derivatives to help with a math test. 2. The derivative of a function $f(x)$ is defined as $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x

1. **State the problem:** We want to find the limit as $h \to 0$ of the difference quotient for $f(x) = 4x + x^2$ at $x=2$, which is $$\lim_{h \to 0} \frac{f(2+h) - f(2)}{h}.$$ Als

1. **State the problem:** Find the slope of the tangent line at $x=2$ for the functions $f(x) = 4x - x^2$ and $g(x) = \frac{1}{3x - 7}$ using the definition of the derivative. 2. *

1. The problem is to simplify or integrate an expression using the substitution $u=9-x^2$. 2. The substitution method involves replacing a complicated expression with a simpler var

1. **State the problem:** We need to evaluate the integral $$\int \frac{x^3}{\sqrt{9 - x^2}} \, dx$$. 2. **Recall the formula and substitution:** When dealing with integrals involv

1. **State the problem:** We need to evaluate the integral $$\int e^{2x} \cos x \left(e^{2x}\right) \, dx$$. 2. **Simplify the integrand:** Notice that the integrand is $$e^{2x} \c

1. **State the problem:** We need to find the integral of the function $e^{2x}$ with respect to $x$. 2. **Formula and rules:** The integral of an exponential function $e^{ax}$ is g

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