∫ calculus

1. **State the problem:** We are asked to find the limit of the piecewise function $$f(x) = \begin{cases} \sin(\pi x) & x > 1 \\ (x-1)^3 & x < 1 \end{cases}$$

1. لنفهم السؤال، لدينا دالة $f(x)$ وقيمة $f(x)$ أكبر من الصفر. 2. المطلوب هو معرفة إذا كان يمكن تعويض $f(x)$ عند نهاية الدالة عندما تقترب $x$ من الصفر من اليمين.

1. **Problem:** Differentiate $y = x^2$. **Formula:** Power rule: $\frac{d}{dx} x^n = n x^{n-1}$.

1. **State the problem:** Solve the integral $$\int \frac{x^2}{\sqrt{16 - x^2}} \, dx$$ using the trigonometric substitution $$x = 4 \sin \theta$$. 2. **Recall the substitution and

1. **Problem Statement:** Solve the integral $$\int \frac{\sqrt{x^2 - 36}}{x} \, dx$$ using the trigonometric substitution $$x = 6 \sec \theta$$. 2. **Formula and Substitution:** W

1. **Problem Statement:** Determine the values of $a$ that make the function

1. مسئله را بیان می‌کنیم: ما یک مستطیل با طول $x$ و عرض $y$ داریم که مساحت آن $A=320000$ است و می‌خواهیم مقدار $x$ و $y$ را پیدا کنیم که محیط (Fence) را به حداقل برساند.

1. Problem 2.1: Given $k(x) = \frac{df}{dx}$, find the constant $C$ such that $\int_4^1 k(x) \, dx = f(4) + C$. Step 1: Recall the Fundamental Theorem of Calculus: If $k(x) = f'(x)

1. We are asked to solve three integrals: $$\int \frac{4x}{x^2 - 3} \, dx$$, $$\int x e^x \, dx$$, and $$\int \ln(x) \, dx$$. 2. For the first integral, use substitution. Let $$u =

1. **Stating the problem:** We need to solve three integrals:

1. **State the problem:** We need to find the derivative of the function $$f(x) = \cos\left(\frac{3x}{2}\right)$$. 2. **Recall the formula:** The derivative of $$\cos(u)$$ with res

1. **State the problem:** We need to evaluate the integral $$\int \frac{x^2}{4 + x^2} \, dx$$. 2. **Rewrite the integrand:** Notice that $$\frac{x^2}{4 + x^2} = \frac{4 + x^2 - 4}{

1. **State the problem:** We want to find the point $x$ on the line between two smokestacks, $d$ miles apart, where the concentration $S$ of soot deposits is minimized. The concent

1. **State the problem:** Find the derivative of the function $$h(x) = 16x^{\frac{5}{2}} - 2x^{\frac{1}{2}}.$$\n\n2. **Recall the power rule for derivatives:** If $$f(x) = x^n,$$ t

1. **Stating the problem:** We want to find the limit as $x \to -\infty$ of the function $$f(x) = x - \sqrt{x^2 - 2}$$

1. السؤال: ما هي المشتقة؟ 2. المشتقة هي مفهوم في الرياضيات يُستخدم لحساب معدل التغير اللحظي لدالة بالنسبة لمتغيرها.

1. لنبدأ بتوضيح المشكلة: لديك تعبير وتريد إيجاد مشتقته. 2. لنفترض أن التعبير هو $$\frac{2x}{x^2} - \frac{1}{x}$$.

1. نبدأ بكتابة الدالة المعطاة: $$y=\ln^2(x)-4\ln(x)$$. 2. نريد إيجاد المشتقة $y'$ بالنسبة لـ $x$.

1. **State the problem:** We need to find the integral $$\int \frac{1}{4 - 9x^2} \, dx$$. 2. **Recognize the form:** The denominator is a difference of squares: $$4 - 9x^2 = (2)^2

1. **State the problem:** Evaluate the limit $$\lim_{t \to \infty} \frac{7t - 12t^2}{\sqrt{t + 6t^2}}.$$\n\n2. **Recall limit laws and strategies:** When $t \to \infty$, dominant t

1. **State the problem:** We are given parametric equations \(x = t^2 + 1\), \(y = 4t - 3\), and \(z = 2(t^2 - 3t)\) and need to find the values of \(x\), \(y\), and \(z\) at \(t =