∫ calculus

1. **Problem Statement:** We will solve multiple integral problems including the fundamental theorem of calculus, average value of functions, definite and indefinite integrals, imp

1. **Problem:** State the fundamental theorem of calculus, and hence evaluate the definite integral $$\int (3x^2 + 6) \, dx$$. 2. **Fundamental Theorem of Calculus:** It states tha

1. **Énoncé du problème :** Trouver une primitive de la fonction $f$ sur l'intervalle $I$ dans chacun des cas suivants. 2. **Rappel de la formule :** La primitive $F$ d'une fonctio

1. **Problem Statement:** We have two functions $f(x) = x^2 - 6$ and $g(x) = x^2 - 5$ defined on the interval $0 \leq x \leq 1$. We want to find the volume of the solid formed by r

1. **State the problem:** We need to find the integral $$\int \arctan\left(\sqrt[3]{6x} - 1\right) \, dx.$$\n\n2. **Recall the formula for integration by parts:** \n$$\int u \, dv

1. **State the problem:** Evaluate the double integral

1. **State the problem:** Evaluate the double integral

1. **Problem:** Find the Maclaurin polynomial for $f(x) = \ln(3 + 4x)$. 2. **Formula and rules:** The Maclaurin polynomial is the Taylor polynomial centered at 0:

1. Soal 1 meminta kita menghitung nilai integral $$\int 132 t^3 \, dt$$. 2. Rumus integral pangkat adalah $$\int t^n \, dt = \frac{t^{n+1}}{n+1} + C$$, di mana $n \neq -1$.

1. Problem: Find the derivative of the function $f(x)=3x^2-5x$ from first principles. 2. Formula: The derivative from first principles is given by $f'(x)=\lim_{h\to 0} \frac{f(x+h)

1. **Problem:** Find the derivative of the function $f(x) = 3x^2 - 5x$ from first principles. 2. **Formula:** The derivative from first principles is given by:

1. **Problem:** Given $y = 3x^4 - 6x^2 + x - 4$, find (i) $y'(2)$ and (ii) $y'''(2)$. 2. **Step 1: Differentiate $y$ to find $y'$**

1. **Problem Statement:** Calculate the integral $\int (3x^2 + 2x + 1) \, dx$. 2. **Formula Used:** The integral of a sum is the sum of the integrals, and the power rule for integr

1. We are asked to find the limit of the function $f(x)$ as $x$ approaches 0. 2. From the graph description, the function has vertical asymptotes at $x = -2$ and $x = 2$, and it is

1. **State the problem:** Find the derivative of the function $$f(x) = -4x^6 + \frac{1}{x^7} - 5x^{\frac{5}{7}}$$. 2. **Recall the derivative rules:**

1. Problema este să calculăm integrala \( \int \frac{1}{x^2 - 4} \, dx \). 2. Observăm că \( x^2 - 4 \) este o diferență de pătrate și se poate factoriza astfel: \( x^2 - 4 = (x -

1. Problema este să calculăm integrala \( \int \frac{1}{x^2 - 44} \, dx \). 2. Formula de bază pentru integrala funcțiilor de forma \( \int \frac{1}{x^2 - a^2} \, dx \) este \( \fr

1. The problem is to find the limit: $$\lim_{x \to +\infty} \frac{e^x}{x^{100}}$$. 2. We use the fact that exponential functions grow faster than any polynomial function as $x$ app

1. **Problem:** Find the limit $$\lim_{x \to 0} \frac{e^x - 1}{\sin x}$$ using L'Hopital's Rule. 2. **Recall L'Hopital's Rule:** If the limit results in an indeterminate form like

1. **State the problem:** We need to evaluate the integral $$\int t^2 \cot t \, dt$$. 2. **Recall the formula and rules:** The cotangent function is $$\cot t = \frac{\cos t}{\sin t

1. **State the problem:** Find the limit $$\lim_{x \to \infty} -x^2 + 7x - 1$$. 2. **Recall the rule for limits at infinity of polynomials:** The term with the highest power domina