∫ calculus

1. **Problem Statement:** State Leibnitz theorem for the nth derivative of the product of two functions and find the nth derivative of (i) $x^3 \sin x$ and (ii) $x^2 e^{2x}$. 2. **

1. **State the problem:** Find the limit $$\lim_{x \to 2} k(x)$$ where $$k(x) = \frac{x^3 - 2x - 4}{x - 2}$$.

1. **State the problem:** We need to evaluate the integral $$\int x^3 e^{x^2} \, dx$$. 2. **Identify the method:** This integral suggests using substitution because of the composit

1. **State the problem:** We need to solve the integral $$\int x^3 e^{x^2} \, dx$$. 2. **Identify the method:** This integral suggests using substitution because of the composite f

1. **Problem:** Evaluate the integral $$\int \frac{3}{4 + t^2} \, dt$$. 2. **Formula:** Recall the integral formula for $$\int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \arctan\left(

1. **State the problem:** We need to find the value of the function $g(x) = \frac{1}{2} e^x (\sin x - \cos x)$ at $x=0$. 2. **Recall the formula:** The function is given by

1. Énoncé du problème : Trouver les fonctions primitives des fonctions données. 2. Rappel de la formule : La primitive de $x^n$ est $\frac{x^{n+1}}{n+1} + C$ pour $n \neq -1$.

1. **Énoncé du problème :** Calculer $I_1 = \int_0^1 x \sqrt[3]{1 - x} \, dx$ en utilisant une intégration par parties, puis montrer que $I_1 = \frac{3}{7} I_0$ où $I_0 = \int_0^1

1. Problem: Find the limits of the following expressions as the variable approaches infinity or negative infinity. 2. Formula and rules: For rational functions, divide numerator an

1. Problem: Find the limits of the given functions as the variable approaches infinity or negative infinity. 2. Formula and rules: For rational functions, divide numerator and deno

1. **Problem:** Find the value of the limit $$\lim_{x \to 3} \frac{x^3 - 27}{x - 3}$$. 2. **Recall the formula:** This is a limit of the form $$\frac{f(x) - f(a)}{x - a}$$ where $$

1. **State the problem:** We need to find the integral $$\int \frac{t^4}{t^3+1} \, dt.$$\n\n2. **Rewrite the integrand:** Notice that the degree of the numerator ($4$) is higher th

1. **Stating the problem:** We want to understand what a derivative is and see some examples. 2. **Definition:** The derivative of a function measures how the function's output cha

1. **Stating the problem:** We need to find the integral $$\int x \sqrt{ax+b} \, dx$$ where $a$ and $b$ are constants. 2. **Formula and substitution:** To solve this integral, we u

1. Diberikan fungsi $f(x) = x^3 + 3x^2 - 9x - 1$. Kita diminta menentukan interval dimana fungsi ini naik dan turun. 2. Untuk menentukan interval naik dan turun, kita gunakan turun

1. **Stating the problem:** We are given a function $$F(x) = -x + (x - 2)\ln(2 - x) + e^{x-1}$$ and asked to evaluate the average value of its derivative $$f'(x) = \frac{n(x)}{x-2}

1. **Problem:** Find the derivative of $f(x) = \sin(x^2 + 3)$. 2. **Formula and rules:** Use the chain rule: if $f(x) = \sin(g(x))$, then $f'(x) = \cos(g(x)) \cdot g'(x)$.

1. **Problem Statement:** We are given the function $f(x) = \frac{2x}{x^2 + 1}$, representing velocity over time $t \geq 0$. We want to find its antiderivative $F(x)$, graph it, an

1. **Problem:** Determine whether the integral \(\int_0^\infty \frac{1}{\sqrt{x} (1 + x)} \, dx\) is convergent or divergent and evaluate it if convergent. 2. **Formula and rules:*

1. نبدأ ببيان المسألة: إذا كانت د س = 2، نريد فهم معنى هذا التعبير. 2. د س عادةً تعني مشتقة دالة س بالنسبة إلى متغير معين، غالبًا x.

1. Énonçons le problème : Trouver la primitive $J = \int \frac{dx}{\sqrt{-x^2 + 2x + 8}}$. 2. Commençons par simplifier l'expression sous la racine. Complétons le carré pour $-x^2