∫ calculus

1. Let's start with a classic related rates problem: "A balloon is rising vertically at a rate of 5 meters per second. A person is walking away from the balloon's launch point at 3

1. **Problem:** Determine if the statement "If f is differentiable on [-1,1] then f is continuous at x = 0" is true. 2. **Recall the theorem:** Differentiability implies continuity

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

1. The problem asks to determine the truth value of the statement: "If $f$ is differentiable on $[-1,1]$ then $f$ is continuous at $x=0$." 2. Recall the important rule: Differentia

1. مسئله را بیان می‌کنیم: محاسبه انتگرال $$\int \frac{(\ln x)^3}{x} \, dx$$. 2. برای حل این انتگرال از جایگذاری استفاده می‌کنیم. فرض کنید $$t = \ln x$$.

1. **State the problem:** We want to find the limit $$\lim_{n \to \infty} \frac{\ln m}{\ln n}$$ where $m$ is a constant and $n$ approaches infinity. 2. **Recall the properties:** T

1. نگاهی به مسئله: باید انتگرال $$\int \frac{x}{1+x^2} \, dx$$ را محاسبه کنیم. 2. فرمول و قانون مهم: برای انتگرال‌هایی که صورت کسر مشتق مخرج یا بخشی از آن است، می‌توان از جایگزینی

1. **State the problem:** We need to evaluate the integral $$\int \frac{m}{1 e^x} \, dm$$. 2. **Analyze the integral:** The integral is with respect to $m$, and the integrand is $$

1. The problem is to find the integral $\int t^2 \cos t \, dt$. 2. We use integration by parts, which states:

1. **State the problem:** We want to understand why $\int \cos 5x \, dx = \frac{1}{5} \sin 5x + C$. 2. **Recall the integral rule:** The integral of $\cos(ax)$ with respect to $x$

1. **Problem Statement:** Find the area bounded by the curves $y = x$ and $y = x^2$. 2. **Formula and Rules:** The area between two curves $y = f(x)$ and $y = g(x)$ from $x=a$ to $

1. **Problem statement:** Calculate the limit $$\lim_{x \to 0^+} (\sin x \cdot \ln x)$$ where $x$ approaches $0$ from the right (since $\ln x$ is defined only for $x>0$). 2. **Reca

1. **State the problem:** We need to evaluate the integral $$\int \frac{dx}{\sec 5x}$$. 2. **Rewrite the integrand:** Recall that $$\sec \theta = \frac{1}{\cos \theta}$$, so $$\fra

1. **بيان المسألة:** نحسب النهاية $$\lim_{x \to 0} (\sin x \cdot \ln x)$$ حيث $x \to 0$ من الجانب الموجب لأن $\ln x$ معرف فقط لـ $x>0$.

1. **Problem:** Find the derivative of the function $$f(x) = 3\sqrt{x} + 4x^6 + 5$$. 2. **Recall the formulas:**

1. **Problem Statement:** Show that the Laplace transform of the piecewise function

1. **Evaluate the limit** $$\lim_{(x,y) \to (0,0)} \frac{xy}{x^2 + y^2}$$ - We check the limit along different paths to see if it exists.

1. The problem is to find the derivative of the function $y = \tan(x)$.\n\n2. The formula for the derivative of the tangent function is $\frac{d}{dx} \tan(x) = \sec^2(x)$.\n\n3. Th

1. Masalah: Hitung luas daerah yang dibatasi oleh kurva $y = x^4 - 4x^2$ dan garis vertikal $x=1$ serta $x=2$ di atas sumbu $x$. 2. Rumus yang digunakan adalah rumus luas daerah di

1. The problem asks to find the partial derivative of the function $f(x,y,t) = x^2 y^3 t^4 - x y^4 t^2 + x^4 t^5$ with respect to $t$, and then evaluate it at the point $(1,1,1)$.

1. Diberikan integral $$\int \frac{6x}{\sqrt{2x^2 + 3}} \, dx$$. Kita diminta menentukan hasil integral ini. 2. Gunakan substitusi untuk menyelesaikan integral ini. Misalkan $$u =