∫ calculus

1. **State the problem:** Find the limit $$\lim_{x \to -\infty} \left(1 + (x^2 - 1)e^{-x}\right).$$\n\n2. **Recall the behavior of terms:** As $x \to -\infty$, $x^2$ grows very lar

1. **Problem 1a:** Differentiate $$f(x) = \frac{3x^2 - 2}{4x^{-3} - 2\sqrt{4\sqrt{x} + 1}}$$ - Use the quotient rule: $$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$$ where $$

1. The problem is to find the limit $$\lim_{x \to -\infty} e^{-x}$$. 2. Recall the exponential function rule: for any real number $a$, $e^a$ grows very fast as $a$ increases.

1. **Problem:** Use integration by parts to find the integral $$\int t \ln(t^2) \, dt$$. 2. **Formula:** Integration by parts formula is $$\int u \, dv = uv - \int v \, du$$.

1. مسئله: حد $$\lim_{x \to 0} \left(\frac{1}{x}\right)^5$$ را با استفاده از قضیه ساندویچ بیابید. 2. قضیه ساندویچ می‌گوید اگر برای تابعی $f(x)$ داشته باشیم که $$g(x) \leq f(x) \leq

1. **Stating the problem:** We have a differential equation $$y' = \frac{14.1x^2 + 3x + 2}{x^3 + x^2}$$ for $$x > 0$$ with initial condition $$y(1) = 6.3$$. We need to evaluate num

1. **Problem statement:** We are given a differential equation $$y' = \frac{10.2x^2 + 15.6x + 2}{x^3 + 3x^2 + 2x}$$ for $$x > 0$$ with initial condition $$y(0.5) = 2.5$$. We need t

1. **State the problem:** We want to find the integral $$\int \frac{ax^3 + bx^2 + cx + d}{ex^2 + fx + g} \, dx$$ where $a,b,c,d,e,f,g$ are constants. 2. **Understand the approach:*

1. مسئله: مشتق تابع $$F(x) = (5x^3 - 2x + 4)^{\frac{9}{4}}$$ را پیدا کنیم. 2. فرمول مورد استفاده: برای مشتق تابعی به صورت $$[g(x)]^n$$ از قاعده زنجیره‌ای استفاده می‌کنیم:

1. **State the problem:** Evaluate the limit $$\lim_{x \to 0} \frac{1 - \cos x + \sin 3x}{1 + \tan 2x - \cos x}$$. 2. **Recall important formulas and approximations near zero:**

1. **State the problem:** Find the reduction formula for $$I_n = \int x^n \cosh^n(ax) \, dx$$. 2. **Recall the formula and rules:** We will use integration by parts and properties

1. Stated problem: Calculate the integral $$\int (x^2 + 5x) \, dx$$. 2. Formula and rules: The integral of a sum is the sum of the integrals, and the power rule for integration sta

1. We are asked to find the integral of the function $x^2 + 5x$ with respect to $x$. 2. The formula for integrating a sum is the sum of the integrals: $$\int (f(x) + g(x)) \, dx =

1. مسئله: حد $\lim_{x\to 1} \frac{x^{n+1} - (n+1)x + n}{(x-1)^2}$ را برای $n\in\mathbb{N}$ محاسبه کنید. 2. روش: برای حل، از جایگذاری $h=x-1$ و بسط دوجمله‌ای استفاده می‌کنیم.

1. **Problem statement:** Find the limit $$\lim_{x \to a} \frac{x^{1/m} - a^{1/m}}{x - a}$$ without using L'Hospital's rule or differentiation. 2. **Recall the formula for differen

1. **State the problem:** Compute the limit $$\lim_{x \to \infty} \frac{5x^4 - 7x + 2}{-2x^4 + 2x - 3}$$. 2. **Recall the rule for limits of rational functions as $x \to \infty$:**

1. Diberikan fungsi $f(x) = x^3 - 3x^2 + 2$. Kita diminta mencari titik-titik kritis dan menentukan apakah titik tersebut maksimum lokal atau minimum lokal menggunakan Uji Turunan

1. **Problem:** Prove that $$\frac{dy}{dx} = x^{\sin x - 1} \left( \sin x + x \cos x \ln x \right)$$ is the derivative of $$y = x^{\sin x}$$. 2. **Formula and rules:** For function

1. The problem is to evaluate the definite integral $$\int_1^8 x \, dx$$. 2. The formula for the integral of a power function is $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ where

1. **Problem Statement:** We are given the graph of $f'$, the derivative of a function $f$, and need to determine the intervals on which $f(x)$ is concave up on the open interval $

1. Diberikan fungsi $f(x) = x\sqrt{x^2 + 1}$. Kita diminta menentukan turunan fungsi tersebut dan membandingkan dengan pernyataan Mahasiswa A dan B. 2. Gunakan aturan turunan hasil