∫ calculus

1. The problem is to understand the concept of differentiation in calculus. 2. Differentiation is the process of finding the derivative of a function, which represents the rate of

1. The problem is to differentiate the function $$f(x) = \frac{(x^2 + 1)^2}{x^2 - 1}$$ with respect to $x$. 2. First, identify the numerator and denominator:

1. **State the problem:** We need to find the exact value of $a$ where the curve $y = \cos x \sqrt{\sin 2x}$ has a maximum point $M$ for $0 \leq x \leq \frac{\pi}{2}$. 2. **Write t

1. **Тодорхойлолт:** Дуу бичлэгийн компани ТВ сурталчилгааны өдрийн тоог $t$ гэж үзвэл, $t$ хоногийн дараах CD худалдан авах хувь нь $$1 - e^{-0.06t}$$ байна.

1. The problem is to find the integral $$\int x^3 \cos(x^4 + 2) \, dx$$. 2. Notice that the argument of the cosine function is $$x^4 + 2$$, and its derivative is $$4x^3$$, which is

1. The problem is to differentiate the function $$f(x) = x^4 + 2x^3 + x^2$$ with respect to $$x$$. 2. Recall the power rule for differentiation: $$\frac{d}{dx} x^n = nx^{n-1}$$.

1. **State the problem:** We want to evaluate the integral $$I = \int_1^4 \frac{\sqrt{x-1}}{2(x+\sqrt{x})} \, dx$$ using the substitution $u = \sqrt{x}$. 2. **Substitution:** Let $

1. The problem is to find the original function $F(x)$ given its derivative $F'(x) = 3 + \frac{5x^2 + 2}{x^{1/2}}$. 2. Rewrite the derivative to simplify the integral:

1. **State the problem:** Evaluate the definite integral $$- \int_0^1 z e^{z^2} \, dz$$. 2. **Rewrite the integral:** The negative sign can be factored out:

1. The problem is to find $\frac{dy}{dx}$ for the parametric equations $y = t^6 - 5$ and $x = 4^3 - 1$. 2. First, note that $x = 4^3 - 1 = 64 - 1 = 63$ is a constant, independent o

1. The problem is to find the derivative $\frac{dy}{dx}$ of a function $y$ with respect to $x$. 2. To proceed, please provide the explicit function $y=f(x)$ you want to differentia

1. **State the problem:** Find the derivative of the implicit function defined by the equation $$2y^3 + 3x^5 = 5y - 6$$ with respect to $x$. 2. **Differentiate both sides with resp

1. The problem asks to differentiate functions using \(dy\) and explain the meaning or purpose of \(d\) in it. 2. In calculus, \(dy\) represents the differential of \(y\), which is

1. **State the problem:** Find the limit of the piecewise function $$f(x)$$ as $$x \to 3$$, where $$f(x) = \begin{cases} 4 - x, & x < 3 \\ 7, & x = 3 \\ x^2 - 8, & x > 3 \end{cases

1. The problem is to find the limit of the piecewise function \(f(x)\) defined as: $$f(x) = \begin{cases} 4 - x & \text{if } x < 3 \\ 7 & \text{if } x = 3 \\ x^2 - 8 & \text{if } x

1. **Problem:** Find the average rate of change of $y=3x^2$ from $x=-1$ to $x=2$. 2. **Step 1:** Calculate $y$ at $x=2$:

1. **State the problem:** We need to evaluate the integral $$\int x \ln x \, dx$$. 2. **Choose a method:** Use integration by parts, where $$\int u \, dv = uv - \int v \, du$$.

1. **State the problem:** We need to evaluate the integral $$\int \cos x \ e^x \, dx$$. 2. **Choose method:** Use integration by parts, where we let:

1. Problem: Rozwiąż całkę nieoznaczoną \( \int f(x) \, dx \) podaną przez użytkownika. 2. Niestety, przesłany tekst zawiera dane binarne i metadane, a nie wyraźne wyrażenie matemat

1. **State the problem:** We want to find the function $f(x)$ defined by the integral $$f(x) = \int \frac{\sqrt{x^2 + 3 - 2}}{x - 1} \, dx, \quad x \neq 1.$$ Simplify the expressio

1. We are given four piecewise functions and asked to determine if each function is continuous at the specified points. 2. For each function, continuity at a point means: