∫ calculus

1. The problem asks to find the arc length of the curve defined by the function $y = x^3$ from $x=0$ to $x=2$. 2. The formula for the arc length $L$ of a function $y = f(x)$ from $

1. The problem is to find the indefinite integral $$\int (12x^6 + 7x^5 + 2)\,dx.$$\n\n2. We use the rule for integrating powers of $x$: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C.$

1. The problem asks us to find the indefinite integral of the function $12x^6 + 7x^5 + 2$ with respect to $x$. 2. We apply the power rule for integration to each term separately. R

1. Statement of the problem: Compute the integral $\int (12x^6+7x^5+2)\,dx$. 2. Reasoning: Use linearity to integrate term-by-term and the power rule $\int x^n\,dx=\frac{x^{n+1}}{n

1. **State the problem:** We need to evaluate the integral

1. Find the four second-order partial derivatives of the given functions. **For i.** $f(x,y) = 2x^2 y^3$

1. **Problem:** Evaluate $$\lim\limits_{(x,y)\to(0,\ln 2)} e^{x-y}$$ Step 1: Substitute $x=0$ and $y=\ln 2$ directly since exponential is continuous.

1. The problem is to find the limit $$\lim_{x\to 1}(5 - 4x)$$. 2. Recall that if the function is continuous at $x=1$, the limit can be found by direct substitution.

1. First, state the problem: Find the limit as $x$ approaches 1 of the function $$\frac{3x^3 + 2x^2 - 3x}{x^3 - 1}$$. 2. Substitute $x = 1$ directly into the expression to check if

1. Problem 6 asks to find the increasing and decreasing intervals for a function with two vertical asymptotes between $x=-2$ and $x=2$. The graph approaches $+\infty$ to the left o

1. **Problem statement:** Determine where the function $$f(x) = \frac{x - 1}{x^2 - 1}$$ is continuous and if possible, extend it to a larger domain continuously. 2. **Analyze the d

1. The problem asks to determine the continuity of a function, but no specific function was provided. 2. To analyze continuity, we first need the explicit mathematical expression o

1. Given the function $$y = 8(5x^2 - 1)(x^2 - 1)$$, we want to find the second derivative $$y''$$. 2. First, expand the function:

1. Stating the problem: Find the third derivative $\frac{d^3y}{dx^3}$ of the function $$y=\frac{x^2+2x}{x-1}.$$\n\n2. Simplify the function first by performing polynomial division:

1. The problem: Find the third derivative ($y'''$) of the function $$y = \frac{x^2 + 2x}{x - 1}.$$\n\n2. To differentiate, let's rewrite $y$ as a quotient and apply the quotient ru

1. The problem is to find the third derivative, $y'''$, of the function $$y = \frac{x-1}{x^2 + 2x}.$$\n\n2. First, simplify the denominator: $$x^2 + 2x = x(x+2).$$\n\n3. Use the qu

1. **State the problem:** We need to find the third derivative $y'''$ of the function $$y=\frac{x^2+2x}{x-1}.$$\n\n2. **Rewrite the function to simplify differentiation:** Use the

1. State the problem: Find the third derivative $y'''$ of the function $y = x^2 - \sqrt[3]{4x}$. 2. Rewrite the function using exponents: $$y = x^2 - (4x)^{1/3}.$$

1. Énoncé du problème : Nous devons calculer $dF$ où $F = \sin(x) y^2$ et $\varphi(t) = (t^2, t)$. Ici, $F$ est une fonction de deux variables $x$ et $y$, et $\varphi$ est une fonc

1. We are asked to evaluate the integral $$\int 4x^2 \sqrt{x^3 - 5} \, dx$$. 2. Let us use substitution. Set $$u = x^3 - 5$$.