∫ calculus

1. نبدأ بكتابة المسألة: نريد حساب التكامل $$\int \frac{1+\sin 2x}{1-\sin 2x} \, dx$$. 2. نستخدم الهوية المثلثية: \(\sin 2x = 2 \sin x \cos x\) لكن هنا سنعالج التعبير كما هو.

1. **State the problem:** We need to find the derivative of the function given implicitly as $$\frac{dt}{dy} = k y_0^{e^{kt}}$$ with respect to $y$. 2. **Rewrite the problem:** The

1. **State the problem:** Differentiate the function $y(t) = y_0 e^{kt}$ with respect to $t$. 2. **Recall the formula:** The derivative of an exponential function $e^{u(t)}$ with r

1. נניח שהבעיה היא למצוא את האסימפטוטות של פונקציה נתונה. 2. אסימפטוטות אופקיות הן ערכי ה-$y$ שאליהם הפונקציה מתקרבת כאשר $x$ שואף לאינסוף או מינוס אינסוף.

1. **State the problem:** Find the limit as $x \to \infty$ of the function $$\frac{x - 25}{x^2 + 1}$$. 2. **Recall the rule for limits of rational functions at infinity:** When the

1. **Stating the problem:** Find the limit as $x$ approaches 5 of the function $$\frac{x^2 - 4x - 5}{x - 5}$$ 2. **Recall the formula and rules:** When direct substitution leads to

1. **State the problem:** We need to find the integral $\int \ln x^2 \, dx$. 2. **Simplify the integrand:** Using the logarithm power rule, $\ln x^2 = 2 \ln x$.

1. **Problem 1: Find the equation of the tangent to the curve $y = x^3 - 5x + 1$ at $x=2$.** 2. The formula for the slope of the tangent line to a curve at a point is the derivativ

1. **Problem Statement:** We need to find all values of $x$ in the open interval $(-9,9)$ where the function $f$ has an inflection point. 2. **Definition and Formula:** An inflecti

1. مسئله: حد راست و چپ تابع قطعه‌ای داده شده را وقتی که $x$ به عدد 2 میل می‌کند، پیدا کنیم. تابع به صورت زیر تعریف شده است:

1. **Problem statement:** Find the derivative $\frac{dy}{dx}$ of the function $$y = \frac{x + 2}{\sqrt{3x + 4}}$$.

1. **Problem Statement:** Find the area of the shaded region bounded by the curves $y=1$, $y=\frac{1}{4}x$, and $y=\frac{1}{36}x^2$. 2. **Understanding the curves:**

1. **State the problem:** We need to find the indefinite integral of the function $12 + 8x - 4x^2$ with respect to $x$. 2. **Recall the integral rules:**

1. **State the problem:** We need to find the average price $\bar{p}$ of the product on the interval $40 \leq x \leq 50$ where the price function is given by $$p = \frac{132000}{49

1. **State the problem:** Evaluate the integral $$\int_1^e x^{x^x} \cdot x^x \left(\ln x + (\ln x)^2 + \frac{1}{x}\right) \, dx$$.

1. Problem: Evaluate the integral \(\int x e^x \, dx\) using integration by parts with \(u = x\) and \(dv = e^x dx\). 2. Formula: Integration by parts states:

1. **Stating the problem:** We are asked to find the left-hand limit $\lim_{x \to 1^-} f(x)$, the right-hand limit $\lim_{x \to 1^+} f(x)$, the overall limit $\lim_{x \to 1} f(x)$,

1. The problem asks to find the left-hand limit, right-hand limit, two-sided limit, and function value at $x = -2$ for the function $f(x)$ based on the graph. 2. Recall the definit

1. **Problem statement:** Given a piecewise linear function $f$ defined by points $(-4,-3)$, $(-1,-2)$, $(1,2)$, $(2,0)$, and $(3,-2)$, define $g(x) = \int_{-4}^x f(t) \, dt$. Find

1. **State the problem:** We have two particles A and B moving along a horizontal line. Particle A's velocity $v(t)$ is given at specific times, and Particle B's position is given

1. The problem asks to explain why the function in 1d is decreasing. 2. A function is decreasing on an interval if for any two points $x_1$ and $x_2$ in that interval, whenever $x_