∫ calculus

1. Problem Übung 7: Berechnen Sie die Integrale. (a) \(\int (1 - 2 e^{-x}) \, dx\)

1. **State the problem:** We need to evaluate the integral $$\int x^2 \sec^2(x) \, dx$$. 2. **Recall the formula and rules:** The integral involves a product of $x^2$ and $\sec^2(x

1. The problem is to find the derivative of the function $$y = \cos(\sin x)$$ with respect to $$x$$. 2. We use the chain rule for differentiation, which states that if $$y = f(g(x)

1. **Stating the problem:** Find the limit \( \lim_{x \to \infty} \left(5 + \frac{1}{x}\right) \). 2. **Formula and rules:** The limit of a sum is the sum of the limits, provided e

1. Stating the problem: We need to find the integral of the function $x^3$ with respect to $x$. 2. Formula used: The integral of $x^n$ with respect to $x$ is given by $$\int x^n \,

1. **State the problem:** Given the function $y = \frac{3x + 10}{x - 2}$ where $x$ and $y$ are positive and vary with time, and the rate of increase of $y$ is twice the rate of dec

1. **State the problem:** Differentiate the function $y = (4x)^{0.5}$ with respect to $x$. 2. **Recall the formula:** The derivative of $x^n$ with respect to $x$ is given by $$\fra

1. **Problem:** Find the limit $$\lim_{x \to \infty} \frac{x}{2x - 3}$$ 2. **Formula and rules:** When finding limits at infinity for rational functions, divide numerator and denom

1. **Problem statement:** Given that $y = \frac{3x + 10}{x - 2}$, where $x$ and $y$ are positive and vary with time, and the rate of increase of $y$ is twice the rate of decrease o

1. **State the problem:** Find the limit $$\lim_{x \to 5} \frac{\frac{1}{s} + \frac{1}{x}}{10 + 2x}$$ where $s$ is a constant and $x$ approaches 5. 2. **Understand the expression:*

1. **State the problem:** Evaluate the integral $$\int_0^\infty \frac{x \ln^2(x) \ln(\ln(x^2))}{x^4 - 1} \, dx$$ and multiply the result by the coefficient $$\frac{32i}{7\pi}$$.

1. Let's understand why a function has maximums at certain points. 2. A maximum point on a function is where the function changes from increasing to decreasing.

1. The problem asks us to determine the nature of critical points of the function $f$ at $x=0$ and $x=6.949$ using the second derivative test. 2. The second derivative test states:

1. The problem asks to determine the nature of the critical points at $x=0$ and $x=6.949$ for the function $f$ given its second derivative $$f''(x) = \sin\left(\frac{x^2}{8}\right)

1. **Problem statement:** We are given the second derivative of a function $f$ as $$f''(x) = x^2 \cos\left(\frac{x^2 + 2x}{6}\right)$$ and asked to find the values of $x$ in the in

1. **State the problem:** We are given the time $t$ in seconds for an object dropped from height $s$ meters to reach the ground, with the formula $$t = \sqrt{\frac{s}{5}}.$$ We nee

1. **State the problem:** Find the rate of change of temperature $T(h) = \frac{60}{h+2}$ with respect to height $h$ at $h=3$ km.

1. **State the problem:** Find the volume $V$ of the solid formed by rotating the region enclosed by the curves $y=5x^2+30$ and $y=40x-5$ about the y-axis. 2. **Find the points of

1. **State the problem:** We want to find the volume $V$ of the solid obtained by rotating the region bounded by $y = x^2$, $y = mx$, and the $x$-axis around the $x$-axis using the

1. **State the problem:** We are given the derivative \( \frac{dy}{dx} = \frac{2x - 6}{x^2 - 2x} \) and asked to find the original function \( y \). 2. **Recall the formula:** To f

1. **State the problem:** We have a model car moving on a circular track centered at (20, 0) with radius 5 feet. The fixed point is at the origin (0, 0). We want to find the rate o