∫ calculus

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

1. **Problem statement:** Find the y-coordinates of points A and B where the curve $x=\frac{2}{3}\sin(3y+\frac{\pi}{4})$ intersects the y-axis, i.e., where $x=0$.