∫ calculus

1. **State the problem:** We want to express the integral $I(a,n+1)$ in terms of $I(a,n)$, where $I(a,n)$ is a function involving parameters $a$ and $n$. 2. **Assume the definition

1. **Stating the problem:** We are given the integral $$I(a,n)=\int_0^1 x^a (1-x)^n dx$$ and the special case $$I(a,0)=\int_0^1 x^a dx$$. We need to use integration by parts to est

1. The problem states two integrals: $$I(a,n)=\int_0^1 x^a (1-x)^n dx$$ and $$I(a,n)=\int_a^b f(x) dx$$. We want to understand or relate these integrals. 2. The first integral $$I(

1. Find the integrals: 1.1 \( \int \cot x \, dx \)

1. **Problem:** Find the limit $$\lim_{x \to 0} \frac{\sin(2x)}{2x}$$. **Step 1:** Recall the standard limit formula $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$.

1. a) Discuss the convergence of exponential series $\sum_{n=0}^\infty \frac{x^n}{(n+1)!}$.\n\nStep 1: State the problem: Determine if the series $\sum_{n=0}^\infty \frac{x^n}{(n+1

1. **הבעיה:** נתונה הפונקציה $f(x)=2x^2 e^{x^2 m}$ כאשר $m \neq 0$. א. מצא את תחום ההגדרה של הפונקציה.

1. 문제 설명: 시각 $t=0$에서 점 $A(1)$에서 출발한 두 점 $P$, $Q$가 수직선 위를 움직입니다. 각 점의 속도는 $v_1(t) = 2t - 6$, $v_2(t) = -3t^2 + 5t$입니다. 두 점의 속도가 같아지는 순간의 $t$를 구하고, 그때 두 점 사이의 거리를 구합니다. 2. 속도가 같아지는 순

1. 문제를 이해하기: 다항함수 $f(x)$가 모든 실수 $x$에 대해 $$x f(x) = 2x^3 - x^2 + 6a + \int_a^x f(t) \, dt$$

1. The problem is to evaluate the limit $$\lim_{x \to +\infty} f(x)$$ where the function or expression is not fully provided. 2. To solve a limit as $x$ approaches $+\infty$, we an

1. The problem is to find the limit of a function as the variable approaches a certain value. 2. The general formula for a limit is $$\lim_{x \to a} f(x) = L$$ where $L$ is the val

1. نبدأ بكتابة الدالة المعطاة: $$g(x)=1+(x^2+x-1)e^{-x}$$ 2. المطلوب هو حساب نهاية الدالة عندما يقترب $x$ من $+\infty$.

1. لنبدأ بتوضيح المسألة: لدينا الدالة $g(x) = 2 + (x-1)e^{-x}$ وهي مشتقة الدالة $f(x)$، أي $f'(x) = g(x)$. 2. المطلوب هو فهم العلاقة بين $f$ و $g$، حيث أن $f'(x) = g(x)$ يعني أن $f

1. نبدأ بكتابة المعادلة المعطاة: $$f(x) = 2x + 1 - x e^{-x}$$ 2. نريد إيجاد معادلة المماس عند النقطة التي فيها $x=1$.

1. نبدأ بتحديد المعادلة المعطاة للمنحنى cf (يرجى توضيح المعادلة إذا كانت متوفرة). 2. نحدد النقطة التي عندها نريد إيجاد معادلة المماس، وهي النقطة ذات الفاصلة 1.

1. The problem asks for the half derivative of the function $y=x$. 2. The half derivative is a fractional derivative of order $\frac{1}{2}$, which generalizes the concept of intege

1. **Stating the problem:** We want to evaluate the integral $$\int \frac{\sin(x-\alpha)}{\sin(x+\alpha)} \, dx.$$\n\n2. **Rewrite the integrand:** Use the sine subtraction and add

1. **Stating the problem:** We want to find the integral $$\int \frac{\sin(x+\alpha)}{\sin(x-\alpha)} \, dx$$ where $\alpha$ is a constant. 2. **Formula and approach:** To solve th

1. **State the problem:** Find the surface area of the solid generated by revolving the curve $y=\sqrt{7+3x^2}$ from $x=0$ to $x=1$ about the y-axis. 2. **Formula for surface area

1. **נתון:** הפונקציה $f(x) = \frac{\ln(x+2)}{x+2}$. 2. **(א) i) מציאת תחום ההגדרה:**

1. **Problem Statement:** Find the limit of the function $$f(x) = \frac{x^2 - 1}{x - 1}$$ as $$x \to 1$$ and determine if the function is continuous at $$x=1$$. 2. **Recall the lim