∫ calculus

1. **State the problem:** Evaluate the integral $$\int \frac{x^4}{\sqrt{4 - x^2}} \, dx.$$\n\n2. **Recall the formula and substitution:** For integrals involving expressions like $

1. **Problem:** Evaluate the integral $$\int \frac{dx}{\sqrt{x} + \sqrt[3]{x}}.$$\n\n2. **Formula and rules:** To integrate expressions involving roots, rewrite roots as fractional

1. **Problem statement:** Given the function $g(x) = (x + 1)e^{ax} + b$ with $a,b \neq 0$, and the curve $(G)$ of $g$ has an inflection point at $x=1$ and a horizontal asymptote $y

1. The problem is to evaluate the definite integral $$\int_{e^2}^{e^3} \frac{1}{x} \, dx$$. 2. Recall the integral formula for $$\int \frac{1}{x} \, dx = \ln|x| + C$$, where $$\ln$

1. نبدأ ببيان المشكلة: لدينا دالة ت(س) معرفة على شكل دالة قطعة (piecewise) كالتالي: $$

1. **Problem statement:** Calculate the area of the region bounded by the curve of the inverse function $f^{-1}$, the line $y=x$, and the vertical lines $x=\alpha$ and $x=0$ by ded

1. **Problem:** Find the derivative of $$y = \frac{x^3}{3} + \frac{x^2}{2} + \frac{x}{4}$$ Formula: Use the power rule $$\frac{d}{dx} x^n = n x^{n-1}$$

1. The problem is to find the derivative $\frac{dy}{dx}$ given the implicit equation $$xy + y^2 = 4.$$ 2. We use implicit differentiation, differentiating both sides with respect t

1. **State the problem:** We need to find the integral $$\int e^{2x} \cos(3x) \, dx$$. 2. **Formula and method:** This is an integral of the form $$\int e^{ax} \cos(bx) \, dx$$. Th

1. Diberikan limit $$\lim_{x \to 2} \frac{\sqrt{ax + 2b} + 8}{x - 2} = 7$$. Kita diminta mencari nilai $$a + b$$. 2. Karena limit ini berbentuk $$\frac{f(x)}{x-2}$$ dan limitnya ad

1. **State the problem:** Calculate the value of $$x = \int_0^\pi \left(3 \sin(t) + \frac{12}{\pi^2} t\right) dt + \det(\ln(e^2))$$

1. **Problem:** Find the first and second derivatives of $f(x) = x^2 + x + 8$. Formula: For $f(x) = x^n$, $f'(x) = nx^{n-1}$ and $f''(x) = n(n-1)x^{n-2}$.

1. **Problem:** Find the derivative of $$y = \frac{x^3}{3} + \frac{x^2}{2} + \frac{x}{4}$$ Formula: Use the power rule $$\frac{d}{dx} x^n = n x^{n-1}$$

1. **Problem:** Find the derivative of $$y = \frac{x^3}{3} + \frac{x^2}{2} + \frac{x}{4}$$ Formula: $$\frac{d}{dx} x^n = n x^{n-1}$$

1. **Problem:** Find the derivative of $$y = \frac{x^3}{3} + \frac{x^2}{2} + \frac{x}{4}$$ **Formula:** Use the power rule $$\frac{d}{dx} x^n = n x^{n-1}$$

1. **Problem:** Find the derivative of $$y = \frac{x^3}{3} + \frac{x^2}{2} + \frac{x}{4}$$ 2. **Formula:** Use the power rule $$\frac{d}{dx} x^n = n x^{n-1}$$ and constant multiple

1. **نص المسألة:** (9) احسب التكامل $$\int س^٥ \times س^٣ \, دس$$.

1. **نص المسألة:** السؤال (9) يطلب حساب التكامل المحدد $$\int_0^5 x^2 \, dx$$.

1. **مشكلة (7): قياس الزاوية التي يصنعها المماس للمنحنى** المطلوب: إيجاد قياس الزاوية التي يصنعها المماس لمنحنى الدالة $$ص = س^3 + 4س + 2$$ عند $$س = \frac{1}{3}$$ مع اتجاه موجب لم

1. **نبدأ بحل السؤال الأول:** السؤال بيقول: قياس الزاوية اللي بيعملها المماس للمنحنى ص = ٤س² + س عند س = \frac{1}{2} مع اتجاه موجب لمحور السينات.

1. نبدأ بحل السؤال الأول: إذا كانت الزاوية التي يصنعها المماس للمنحنى $y = x^4 + 3x^2$ مع الاتجاه الموجب لمحور السينات عند $x = \frac{1}{2}$ تساوي كم؟ 2. نعرف أن ميل المماس عند نقط