∫ calculus

1. **Problem:** Prove the operator property: $$F(D^2) \cos(ax + b) = F(-a^2) \cos(ax + b)$$ where $a$ and $b$ are constants. 2. **Formula and rules:** The operator $D$ represents d

1. Diberikan deret Maclaurin untuk fungsi $\cos(x)$: $$\cos(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$$

1. **Stating the problem:** We need to evaluate the limit $$\lim_{x \to +\infty} (-3x^2 + 2\ln x - 2026)$$ and the definite integral $$\int_1^1 \left(\frac{1}{x} - 1\right) dx$$.

1. **Stating the problem:** We need to find the following limits and evaluate the definite integral:

1. **Problem:** Determine if $$\lim_{(x,y) \to (1,1)} \frac{xy}{xy}$$ exists. 2. **Formula and rules:** The limit of a function $$f(x,y)$$ as $$(x,y) \to (a,b)$$ exists if the valu

1. The problem involves evaluating the triple integral $$\int \int \int xe^{-y} \, dy \, dx \, dz$$ over the given limits. 2. The limits are given as:

1. **State the problem:** Find the total area of the region bounded by the curve $f(x) = \sqrt{x^2 - 2x - 4}$, the x-axis, and the vertical lines $x=2$ and $x=10$. The zero of the

1. **State the problem:** We want to find the indefinite integral $$\int \sqrt{\tan x} \, dx$$. 2. **Recall the formula and substitution:** This integral is not straightforward. We

1. **State the problem:** We need to find the integral $$\int \sqrt{\tan x} \, dx$$. 2. **Recall the formula and substitution:** To solve integrals involving square roots of trigon

1. Problem statement: You asked me to solve an integral problem, but no specific integral was provided.

1. **Stel het probleem vast:** We willen bepalen of de functie $$f(x) = \frac{1}{2}x - \sqrt{x}$$ continu en afleidbaar is op het interval $$[0,4]$$. 2. **Definitie continuïteit:**

1. **Stel het probleem vast:** We willen begrijpen wat de Leibniz-notatie is en hoe je de afgeleide van een functie kunt berekenen en interpreteren. 2. **Wat is Leibniz-notatie?**

1. **Problem Statement:** Evaluate the integral $$\int x e^{x^2} \, dx$$. 2. **Choosing the Technique:** This integral suggests using substitution because the exponent $x^2$ is a f

1. **Problem Statement:** Evaluate the limit $$\lim_{x \to 0} \frac{\sin x - x}{x^3}$$ using L'Hôpital's rule. 2. **Recall the formula and rules:** When a limit results in an indet

1. **Problem statement:** Given that $f$ is an odd function, and that $$\int_0^4 f(|x|) \, dx = 1.6,$$ we need to find:

1. **Problem statement:** A particle moves in a straight line with velocity given by $$v(t) = 1 + e^{-t} - e^{-\\sin(2t)}$$ for $$0 \leq t \leq 2$$. (a) Find the velocity at $$t=2$

1. The problem is to identify whether a given series is a p-series. 2. A p-series is a series of the form $$\sum_{n=1}^\infty \frac{1}{n^p}$$ where $p$ is a positive constant.

1. **State the problem:** We want to analyze the convergence of the infinite series $$\sum_{n=r}^\infty \frac{(n-r)!}{n!}$$ where $r$ is an integer. 2. **Rewrite the general term:*

1. **State the problem:** Find the interval of convergence of the series $$1 + \frac{2}{3}x + \frac{4}{9}x^2 + \cdots$$ 2. **Identify the general term:** The series can be written

1. Problem: Find the first derivatives of the given functions. 2. Formula: Use the product rule $\frac{d}{dx}[uv] = u'v + uv'$, chain rule $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$,

1. The problem is to find the integral of the function $y^{-2}$ with respect to $y$. 2. Recall the power rule for integration: $$\int y^n \, dy = \frac{y^{n+1}}{n+1} + C \quad \tex