∫ calculus

1. The problem involves understanding and applying double and triple integrals, including changing the order of integration and variables, and using double integrals to find areas

1. نبدأ بكتابة الدالة المعطاة: $$f(x) = 2\ln(x) - 1 - \frac{1}{x^2}$$ 2. نذكر قواعد الاشتقاق المهمة:

1. The problem is to find the sine series expansion of the piecewise function $$f(x) = \begin{cases} x, & 0 < x < 4 \\ 8 - x, & 4 < x < 8 \end{cases}$$ defined on the interval $(0,

1. **State the problem:** Evaluate the integral $$\int \frac{1}{x^2 \sqrt{4x^2 - 9}} \, dx$$. 2. **Identify the form and substitution:** The integrand contains a square root of the

1. **Problem Statement:** Show that the function $y = x^2 - 6x + 4$ is increasing at $x = 8$, decreasing at $x = 1$, and stationary at $x = 3$. 2. **Formula and Rules:** To determi

1. **Problem:** Show that $f(x) = \cos^2 x$ is decreasing on $(0, \frac{\pi}{2})$. 2. **Formula and rules:** To determine if a function is increasing or decreasing, we use the firs

1. Diberikan fungsi kuadratik $$f(x) = -x^{2} + (c - b)x - bc$$ dengan nilai $a=9$, $b=7$, dan $c=1$ dari NIM 179. 2. Kita diminta menghitung luas daerah di bawah kurva $f(x)$ pada

1. **Problem (a):** Evaluate $$\int_{-1}^0 \int_0^1 (x - y^2) \, dx \, dy$$. 2. **Formula and rules:** For iterated integrals, integrate the inner integral first with respect to $x

1. The problem is to differentiate a function using the chain rule. 2. The chain rule formula is: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$ where $y$ is a function of $

1. **Problem:** Differentiate the following functions with respect to $x$: d) $y = (2x^2 - 5)^9$

1. **State the problem:** Find the volume of the region bounded by the cylinder $x^2 + y^2 = 2y$ and the planes $z + y = 8$ and $z = 2y$. 2. **Rewrite the cylinder equation:** The

1. مسئله: تابع داده شده $y = xe^x + e^x$ است و می‌خواهیم مشتق آن را بررسی کنیم. 2. ابتدا تابع را به دو قسمت تقسیم می‌کنیم: $y_1 = xe^x + xe^x + e^x$ و $y_2 = xe^x$.

1. **Problem statement:** Find the derivative of the function $y = \ln(2 + \sin x)$. 2. **Recall formulas and rules:**

1. **State the problem:** We need to evaluate the definite integral $$\int_1^2 \frac{4x^2 + 2x - 5}{(x + 1)(2x - 1)} \, dx.$$\n\n2. **Rewrite the integrand:** The denominator is al

1. **Problem statement:** Find the derivative of the function \(y = e^{4 \sin x} + \ln(2 + \sin x) + \cosh(\sin x)\). 2. **Recall formulas and rules:**

1. **State the problem:** We need to evaluate the double integral $$\int_{-2}^{2} \int_{x^2}^{2} \cos\left(\sqrt{y^3}\right) \, dy \, dx.$$ This integral is over the region where $

1. The problem is to find the derivative of the function $f(x) = x^3$ with respect to $x$. 2. The formula for the derivative of a power function $x^n$ is given by the power rule:

1. **Problem statement:** Use the Mean Value Theorem (MVT) to show that $$\cos b - \cos a \leq b - a$$ for all real numbers $$a,b$$ such that $$a < b$$. 2. **Recall the Mean Value

1. **Problem Statement:** Find the derivative of the function $$y = (x^2 + 1) \cdot \frac{1}{x}$$. 2. **Formula Used:** Use the product rule for derivatives: $$\frac{d}{dx}[u \cdot

1. **State the problem:** We have the temperature function along a metal rod:

1. **Problem statement:** The temperature along a metal rod is given by a function $T(x)$ where $x$ is the position in meters along the rod from 0 to 62. 2. **Find stationary point