∫ calculus

1. Let's clarify the problem: You are asking about an integral where the limits are from 1 to 5, and you mention it is "on the y" which suggests the integral might be with respect

1. **State the problem:** We need to find the volume of the solid formed when the region enclosed by the curves $y = x^2 + 1$ and $y = 2x + 1$ is rotated about the y-axis. 2. **Fin

1. **Problem:** Find the limit $$\lim_{x \to 1} \frac{\ln x}{x - 1}$$. 2. **Formula and rules:** This is an indeterminate form of type $$\frac{0}{0}$$ as $$\ln 1 = 0$$ and $$1 - 1

1. The problem asks to find the first derivative $y'$ of the function $y = x(\ln x - 1)$.\n\n2. Use the product rule for differentiation: if $y = u v$, then $y' = u' v + u v'$. Her

1. **Problem statement:** Calculate the line integral of the function $f(x,y,z) = x + \sqrt{y - z^2}$ along two curves from $(0,0,0)$ to $(1,1,1)$. 2. **Formula:** The line integra

1. **State the problem:** We are given a function $f$ defined on intervals $[0,8]$ and $[2,8]$ with properties that satisfy Rolle's Theorem and the Mean Value Theorem (MVT). We nee

1. **State the problem:** Find the limit $$\lim_{x\to 0} \frac{\cos 4x}{\tan x}$$. 2. **Recall important formulas and rules:**

1. **State the problem:** Evaluate the improper integral $$I = \int_{e}^{\infty} \frac{dx}{x(\ln x)^3}$$. 2. **Recall the formula and substitution:** For integrals involving $\ln x

1. مسئله: حد تابع $$\lim_{x \to 1} \frac{\sqrt{x+3} - 2}{x - 1}$$ را پیدا کنید. 2. فرمول و قانون: این نوع حدها که به صورت $$\frac{0}{0}$$ هستند، می‌توان با ضرب صورت و مخرج در مزدوج

1. **State the problem:** Differentiate the function $$y = \frac{(x+1)(x-2)^3}{x-3}$$ using logarithmic differentiation. 2. **Recall the formula and rules:** Logarithmic differenti

1. The problem is to evaluate the integral $$\int \frac{6x}{e^{2x}+1} \, dx.$$\n\n2. We will use substitution to solve this integral. Let $$u = e^{2x} + 1.$$ Then, differentiate $$

1. The problem is to find $\frac{dy}{dx}$ given the differential equation $\frac{dy}{dx} = x \ln y$. 2. This is a separable differential equation. We can rewrite it as $\frac{dy}{d

1. **State the problem:** Find the limit $$\lim_{x \to +\infty} \frac{2x-3}{x^2+x+1}$$. 2. **Recall the rule for limits at infinity:** When evaluating limits of rational functions

1. The problem is to solve the differential equation $$\frac{dy}{dx} = x + y$$. 2. This is a first-order linear differential equation. The standard form is $$\frac{dy}{dx} - y = x$

1. Problem: Evaluate $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \int_0^{2\cos\theta} d r d \theta$$. 2. Inner integral: $$\int_0^{2\cos\theta} d r = r \Big|_0^{2\cos\theta} = 2\cos\th

1. The problem is to find the integral of the function $e^x \sin x$, i.e., compute $$\int e^x \sin x \, dx.$$\n\n2. We use the method of integration by parts, which states: $$\int

1. The problem is to evaluate the double integral $$\int_0^1 \int_1^2 x(x+1) \, dy \, dx.$$ 2. The integral is over $y$ from 1 to 2 and $x$ from 0 to 1. The integrand is $x(x+1)$,

1. **State the problem:** Evaluate the definite integral $$\int_0^1 x^7 \sqrt{1 + x^4} \, dx$$. 2. **Identify a substitution:** Let $$u = 1 + x^4$$. Then, $$\frac{du}{dx} = 4x^3$$,

1. The problem asks why the formula for the volume of a sphere is $\frac{4}{3}\pi r^3$ even though a sphere can be thought of as made by adding infinitely many circles. 2. First, l

1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{x-1}{x^2 + x - 2}$$. 2. **Recall the formula and rules:** The limit of a quotient is the quotient of the limits, pro

1. Diberikan fungsi $f(x) = 2x^{2} + 3x + 5$, kita diminta menghitung turunan pertama pada $x=2$ dengan $h=0,1$ menggunakan tiga metode: beda maju, beda mundur, dan beda pusat. 2.