∫ calculus

1. **State the problem:** We are given the function $f(x) = x^3 + 3x^2 - 4x$ with roots at $x = -4, 0, 1$. We need to find the maximum and minimum points of this function and sketc

1. Enunciado del problema: Calcular la integral $$\int t \arcsin(t) \, dt$$. 2. Para resolver esta integral, usaremos integración por partes. Recordemos la fórmula:

1. **State the problem:** We are given the function $$y=\frac{1}{x}$$ and asked to find the open intervals where the function is increasing, decreasing, or constant. 2. **Analyze t

1. **State the problem:** Given parametric equations $x = k(t - \sin t)$ and $y = k(1 - \cos t)$ with $k \neq 0$, find the second derivative $\frac{d^2 y}{dx^2}$ at $t = \frac{\pi}

1. **State the problem:** Evaluate the limit $$\lim_{x \to \infty} \frac{8x^3 - 7x^2 + 2x + 1}{2x^3 + 5x^2 - x - 1}$$. 2. **Identify the highest power of $x$ in numerator and denom

1. The problem asks to find $\frac{dx}{dt}$ when $x = 7 \ln t$. 2. Recall that the derivative of $\ln t$ with respect to $t$ is $\frac{1}{t}$.

1. The problem is to find the derivative of a function, but the function is not specified. 2. To find the derivative, we need the function expression, for example, if the function

1. **Problem a)**: Evaluate or describe the region of integration for $$\int_0^4 \int_{\frac{4-y}{7}}^1 f(x,y) \, dx \, dy.$$

1. **State the problem:** Evaluate the definite integral $$\int_0^\pi \sin^2(t) \cos(t) \, dt$$. 2. **Use substitution:** Let $$u = \sin(t)$$, then $$du = \cos(t) dt$$.

1. Given integrals: \(\int_0^4 x^3 dx = 60\), \(\int_2^4 x dx = 6\). 2. (a) \(\int_0^2 x^2 dx = \left[ \frac{x^3}{3} \right]_0^2 = \frac{8}{3} \approx 2.6667\).

1. The problem is to find the indefinite integral of the polynomial function $$3x^2 + 7x - 2$$ with respect to $$x$$. 2. Recall the power rule for integration: $$\int x^n \, dx = \

1. The problem is to solve the differential equation $$\frac{dy}{dx} = 5y^2 \cos(x)$$. 2. This is a separable differential equation. We can rewrite it as:

1. The problem asks to find the value of $f(0)$ given the derivative $f'(x) = -10x^4 + 8x^3$ and the value $f(1) = 9$. 2. To find $f(0)$, we first need to find the original functio

1. The problem is to solve the differential equation $$\frac{dy}{dx} = 3x^{-2} e^{-y}$$ for $y$ as a function of $x$. 2. Separate variables to isolate $y$ terms on one side and $x$

1. Determine o domínio da função $$z = \frac{1}{\sqrt{1 - x^2 - y^2}}$$. - Para que a função esteja definida, o denominador não pode ser zero e o radicando deve ser positivo.

1. Problem: Find $$\lim_{x \to 16} \frac{x - \sqrt{x-12}}{x - 16}$$. Step 1: Direct substitution gives $$\frac{16 - \sqrt{16-12}}{16 - 16} = \frac{16 - 2}{0} = \frac{14}{0}$$ which

1. Find $$\lim_{x \to 16} \frac{x - \sqrt{x-12}}{x-16}$$ - Direct substitution gives $$\frac{16 - \sqrt{16-12}}{16-16} = \frac{16 - 2}{0} = \frac{14}{0}$$ which is undefined.

1. **Problem:** Find $$\lim_{x \to 16} \frac{x - \sqrt{x - 12}}{x - 16}$$ Step 1: Direct substitution gives $$\frac{16 - \sqrt{16 - 12}}{16 - 16} = \frac{16 - 2}{0} = \frac{14}{0}$

1. The problem is to find the derivative of the function, but the function itself was not provided. 2. To proceed, please provide the explicit function expression you want to diffe

1. The problem is to find the derivative of the square root function, which is $f(x) = \sqrt{x}$.\n\n2. Recall that the square root function can be rewritten using exponents as $f(

1. The problem is to find the limit $$\lim_{x \to 9} \frac{x - 9}{\sqrt{x} - 3}$$. 2. Direct substitution of $x = 9$ gives $$\frac{9 - 9}{\sqrt{9} - 3} = \frac{0}{0}$$, which is an