∫ calculus

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

1. The problem asks to find the limit of the function $f(x)$ as $x$ approaches $-3$. 2. The limit $\lim_{x \to -3} f(x)$ means we want to find the value that $f(x)$ approaches as $

1. **State the problem:** Find the derivative $h'(x)$ of the function $$h(x) = \frac{1}{60}x^2 - \frac{1}{4}x + \frac{3}{5}$$

1. **State the problem:** We are given a differentiable function $h$ with $h(-5)=10$ and its derivative $h'(x)=2-\sqrt{e^x+2x^2}$. We need to find $h(1)$. 2. **Formula used:** Sinc

1. The problem asks for the average value of the function $$g(x)=4\cos\left(\sqrt{x^2+x+5}\right)$$ on the interval $$[0,5]$$. 2. The formula for the average value of a continuous

1. **State the problem:** We need to sketch a function $f$ with the following limit and value conditions: - $\lim_{x \to 2^+} f(x) = 3$

1. The problem asks for the average value of the function $$f(x) = e^{x^2 - 2x}$$ on the interval $$[-1, 3]$$. 2. The formula for the average value of a function $$f(x)$$ on $$[a,b

1. **State the problem:** We need to find the sum of the areas enclosed by the graphs of $$f(x)=2\cos\left(\frac{\pi x}{4}\right)$$

1. **State the problem:** We want to find how many people left the auditorium between minutes $1$ and $5$ given the rate of change of people remaining is $r(t) = -0.1^t$ people per

1. **State the problem:** Find the area of the region between the graphs of $$f(x)=\frac{4}{x}$$ and $$g(x)=5$$ from $$x=-6$$ to $$x=-2$$. 2. **Formula and rules:** The area betwee

1. The problem is to find the derivative $\frac{dy}{dx}$ of a function $y$ with respect to $x$. 2. To solve this, we need the explicit function $y=f(x)$ or an implicit relation inv

1. مسئله: معادله خط مماس بر نمودار تابع $$f(x) = \sqrt{\frac{x+1}{\sqrt{x}-2}}$$ را در نقطه‌ای که طول مماس برابر 4 است، بیابید. 2. ابتدا باید مشتق تابع $$f(x)$$ را پیدا کنیم تا شیب

1. مسئله: نقطه A روی منحنی $y=3 - x^2$ در ناحیه اول دستگاه مختصات قرار دارد. باید مختصات A را طوری پیدا کنیم که مساحت مثلث OAB بیشینه شود. 2. فرمول مساحت مثلث OAB: مثلث OAB با رئوس

1. مسئله: بررسی مشتق‌پذیری تابع $$f(x) = \begin{cases} x^2 + 3 & x \geq 1 \\ 4x & x < 1 \end{cases}$$

1. The problem asks for the derivative of the function $$y = \ln(x + 3) + \ln 5$$. 2. Recall the derivative rule for natural logarithm: $$\frac{d}{dx}[\ln u] = \frac{1}{u} \cdot \f

1. **Problem Statement:** Given the graph of the derivative function $f'(x)$, classify two different key features on $f'(x)$ and determine the corresponding key features on the ori