∫ calculus

1. **State the problem:** We are given a function defined by an integral: $$f(x) = \int (x + 1)(2x^2 + 4x - 1) \, dx$$ with the initial condition $$f(-2) = 1$$. We need to find the

1. The problem asks us to identify which graph represents the curve of the second derivative $f''(x)$ given the graph of the first derivative $f'(x)$. 2. Recall that $f''(x)$ is th

1. **State the problem:** Given the differential equation $$\frac{dy}{dx} = \csc^2 x$$ with the initial condition $$y\left(\frac{\pi}{4}\right) = 2$$, find the value of $$y$$ when

1. **Problem statement:** Calculate the definite integral $$\int_0^{20 \pi} |\sin x| \, dx$$. 2. **Formula and important rules:** The integral of the absolute value of sine over on

1. **Stating the problem:** We are given the integral $$\pi \int_{-2}^2 (4 - x^2) \, dx$$ and asked to identify which solid's volume it represents. 2. **Understanding the integral:

1. **Problem 2:** Find the equation of the tangent line to the graph of a function at the point $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$. To find the tangent line, we

1. **State the problem:** We need to find the definite integral $$\int_{-1}^{4} f(x) \, dx$$ where $$f(x) = \begin{cases} 2x - 1 & \text{for } -1 \leq x \leq 2 \\ 3 & \text{for } 2

1. **Problem statement:** Given a function $f$ defined on $\mathbb{R}$ such that $f(-x) = f(x)$ (meaning $f$ is an even function), and the integrals $\int_{-5}^1 f(x)\,dx = 24$ and

1. **Stating the problem:** We are given the equation $$2\int_0^{10} f(x) \, dx + 8\int_0^{11} f(x) \, dx + 10\int_0^{8} f(x) \, dx = 9$$ and need to find the value of $$2\int_0^{1

1. The problem asks to evaluate the definite integral $$\int_1^e \ln(x) \, dx$$ and choose the correct answer from the options (a) 1/e, (b) e, (c) 1, (d) -1. 2. The formula for int

1. **State the problem:** We are asked to find the limits of the function $g(x)$ as $x$ approaches positive and negative infinity, and to identify the horizontal asymptote(s) of $g

1. **Problem Statement:** We are given a function $f(x)$ with vertical asymptotes near $x = -2$ and $x = 2$, and a horizontal asymptote at $y = 5$. We need to find the limits $\lim

1. The problem states that $$\lim_{x \to \infty} h(x) = 2$$ and asks about $$\lim_{x \to -\infty} h(x)$$ and the horizontal asymptotes of the function $$h(x)$$. 2. Recall that a ho

1. **Problem Statement:** We are given the function $m(q) = 2q e^{-q^2}$ and asked to:

1. **Problem 1:** Find the values of $h$, $m$, and $c$ for the function $f(q) = hq^2 + mq + c$ given the gradient function $4q + 8$ and a stationary value of $-3$. 2. **Problem 2:*

1. **State the problem:** Find the derivative of the function $$y = x \cos^{-1} x - \sqrt{1 - x^2}$$. 2. **Recall formulas and rules:**

1. **State the problem:** Find the derivative of the function $$f(x) = x \ln(\arctan x)$$. 2. **Recall the formula:** To differentiate a product of two functions, use the product r

1. **Problem Statement:** Find the Maclaurin series expansion for the function $f(x) = \frac{1}{x-2}$. 2. **Recall the Maclaurin series formula:** The Maclaurin series of a functio

1. **State the problem:** Given the function $u = e^{2\sin x} \sin y$, find the partial derivatives $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$.\n\n2. **Rec

1. The problem asks whether the estimate from part (a) is an underestimate or an overestimate and to provide a reason. 2. Typically, when estimating values using methods like linea

1. **Problem Statement:** Find the global maximum and minimum of the function $$f(x) = x^3 - 3x$$ on the domain $$D = [-4, -2]$$. 2. **Function and Domain:** The function is a cubi