∫ calculus

1. **State the problem:** We want to evaluate the integral $$\int \frac{1}{x^4 + x^2 + 1} \, dx.$$\n\n2. **Rewrite the integrand:** Notice that the denominator can be seen as a qua

1. সমস্যাটি হলো: $f(x) = x + \frac{1}{x}$ এর জন্য $\delta y$ এবং $dy$ নির্ণয় করতে হবে যখন $x=1$ এবং $\delta x = dx = 1$। 2. প্রথমে, $dy$ নির্ণয়ের জন্য আমরা $f(x)$ এর ডেরিভেটিভ $f

1. **State the problem:** Find the Laplace Transform (L.T.) of the function $f(t) = t e^{2t} \sin t \cos 3t$. 2. **Recall the Laplace Transform definition:**

1. **Evaluate** $$\lim_{x \to 2} \frac{x^2 + 4}{\sqrt{x+2} - \sqrt{3x-2}}$$ Step 1: Substitute $x=2$ directly:

1. مسئله: نشان دهید برای هر $x > a$ داریم $$f(x) > f^{-1}(x)$$ با فرض اینکه برای هر $x \geq a$ داریم $1 > f'(x)$ و $f(a) = a$.\n\n2. ابتدا یادآوری می‌کنیم که $f^{-1}$ تابع معکوس $f

1. **Problem Statement:** Calculate the integral $$\int \frac{1}{(x-1)(x+2)} \, dx$$.

1. Masalah: Nyatakan formulasi matematik untuk mencari isipadu air yang mengalir melalui sungai dari $t=0$ hingga $t=10$ jam menggunakan integral. 2. Formula: Isipadu air yang meng

1. **State the problem:** Find the area of the region bounded by the curves $y = x^2 - 2$ and $y = 2$ by finding their intersection points and integrating. 2. **Find the intersecti

1. Masalah ini meminta kita mencari isipadu air yang mengalir melalui sungai dari masa $t=0$ hingga $t=10$ jam. 2. Fungsi kadar aliran sungai diberikan oleh $$Q(t) = 100te^{-0.2t}$

1. Let's start by stating the problem: How to solve equations involving limits as the variable approaches infinity. 2. The key formula is the definition of limit at infinity: $$\li

1. **Problem:** Find the volume under the surface $z = 3x^3 + 3x^2 y$ over the rectangle $R = \{(x,y): 1 \leq x \leq 3, 0 \leq y \leq 2\}$. 2. **Formula:** The volume under a surfa

1. **Stating the problem:** Find the area of the region bounded by the curve $y = x^2 - 3x$ and the line $y = 3 - x$. 2. **Find the points of intersection:** Set the two equations

1. The problem is to find the partial derivative of $\cos x$ with respect to $x$. 2. The formula for the derivative of $\cos x$ is $\frac{d}{dx} \cos x = -\sin x$.

1. The problem is to simplify the expression $v\dot{a^2b} + ab^2$. 2. First, clarify the notation: $v\dot{a^2b}$ likely means $v$ times the derivative of $a^2b$ with respect to som

1. We are asked to evaluate the definite integral $$\int_0^1 \frac{x^5}{x^3 + 2} \, dx.$$\n\n2. The integral involves a rational function where the numerator and denominator are po

1. **Problem statement:** Evaluate the double integral $$\iint_R \frac{xy}{x^2 + 1} \, dA$$ where $$R = \{(x,y) \mid -1 \leq x \leq 1, 0 \leq y \leq 1\}$$ using symmetry. 2. **Unde

1. **State the problem:** Evaluate the limit $$\lim_{x \to 5} \frac{\sqrt{5} - \sqrt{x}}{x - 5}$$. 2. **Recall the formula and rules:** Direct substitution gives $$\frac{\sqrt{5} -

1. **State the problem:** Find the limit $$\lim_{x \to 5} \frac{\sqrt{5} - \sqrt{x}}{x - 5}$$. 2. **Recall the formula and rules:** Direct substitution gives $$\frac{\sqrt{5} - \sq

1. **State the problem:** Given $z = \log(x+y)$, with $x = 3u - v$ and $y = \frac{u^2}{2}$, find the partial derivatives $\frac{\partial z}{\partial u}$ and $\frac{\partial z}{\par

1. **State the problem:** Given $$z = e^x \sin y$$ with $$x = s t^2$$ and $$y = s^2 t$$, find the partial derivatives $$\frac{\partial z}{\partial s}$$ and $$\frac{\partial z}{\par

1. **Problem Statement:** Find the values of $a$ and $k$ given that the derivative of $2ax^7 + 3x^k$ is $42x^6 + 15x^{k - 1}$.