∫ calculus

1. **Stating the problem:** Evaluate the definite integral

1. **State the problem:** Evaluate the integral $$\int \frac{x^3 + 4}{x^2} \, dx.$$\n\n2. **Rewrite the integrand:** Simplify the expression inside the integral by dividing each te

1. נניח את הבעיה: לחשב את הגבול $$\lim_{x \to \infty} \frac{3x(x+1)^4}{(2x-1)^5}$$. 2. נשתמש בכלל הגבולות לפונקציות פולינומיות וחזקות: כאשר $x \to \infty$, הביטוי שגדל הכי מהר בדומ

1. **Problem:** Evaluate $\frac{dy}{dx}$ when $1 + xy^2 + x^2y = 0$. 2. **Formula and rules:** Use implicit differentiation. Differentiate both sides of the equation with respect t

1. **Stating the problem:** We are given the acceleration function $$a(t) = 0.012 + 0.008 \sin(0.5 t)$$ and need to find the velocity $$v(t) = \int_0^t a(s) \, ds$$ and the positio

1. **Problem Statement:** We have a spherical balloon with radius $r$ increasing at a constant rate of $\frac{dr}{dt} = 2$ cm/sec.

1. **State the problem:** The height $h$ of a cone is increasing at a rate of $\frac{dh}{dt} = 10$ cm/sec, and the radius $r$ is changing so that the volume $V$ remains constant. W

1. The problem is to understand the limit expression for the derivative: $$f'(x) = \lim_{h \to 0} \frac{12xh + 6h^2}{h}$$

1. Evaluate the integral $$\int e^x \cos 2x \, dx$$ Step 1. State the problem: We want to find $$\int e^x \cos 2x \, dx$$.

1. **State the problem:** We want to find the limit $$\lim_{x \to 0} \frac{\sin x - \sin x \cos x}{x^3}$$. 2. **Rewrite the expression:** Factor out $\sin x$ in the numerator:

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{\cos 2x - \cos x}{1 - \cos x}$$. 2. **Recall relevant formulas and rules:**

1. The problem is to find the derivative of the function $f(x) = x^2 + 3x$. 2. The derivative of a function $f(x)$, denoted $f'(x)$, represents the rate of change or slope of the f

1. مسئله را بیان می‌کنیم: می‌خواهیم حد عبارت $$\lim_{n \to 3} \frac{\sqrt{5n + 1}}{3n - n^2}$$ را محاسبه کنیم. 2. ابتدا مقدار تابع را در نقطه $n=3$ جایگذاری می‌کنیم تا ببینیم آیا م

1. **Define Homogeneous Function:** A function $f(x,y)$ is called homogeneous of degree $n$ if for all $t > 0$, it satisfies $$f(tx, ty) = t^n f(x,y).$$

1. Problem: Find $\frac{dy}{dx}$ if $f(x,y) = c$ (constant). Formula: For implicit functions, $\frac{dy}{dx} = -\frac{f_x}{f_y}$ where $f_x = \frac{\partial f}{\partial x}$ and $f_

1. **State the problem:** Calculate the limit $$\lim_{n \to \infty} \frac{3 + (-3)^n}{4^n}$$ by definition. 2. **Recall the limit definition and properties:** For sequences, if the

1. **Problem Statement:** Find the derivatives of all trigonometric functions including simple, inverse, hyperbolic, and inverse hyperbolic functions. 2. **Formulas and Rules:** Th

1. **State the problem:** We need to find the limit $$\lim_{x \to 1^+} (x^2 - 1) \tan \frac{\pi x}{2}$$. 2. **Recall the formula and behavior:** As $x \to 1^+$, note that $x^2 - 1

1. **Problem:** Find the limit $$\lim_{x \to 0} \frac{x - \sin x}{x^3}$$ without using L'Hospital's rule. 2. **Recall the Taylor series expansion:** For small $x$, $$\sin x = x - \

1. **Problem statement:** Evaluate the line integral $$\int_C yz\,dx - xz\,dy + xy\,dz$$ where the curve $C$ is given by the parametric equations $x=\sin t$, $y=\cos t$, $z=t^2$ fo

1. **Stating the problem:** We want to evaluate the expression $S = Syzdx - xzdy + xydz$ where $x = \sin t$, $y = \cos t$, and $z = t^2$ for $0 \leq t \leq \frac{\pi}{2}$. The prob