∫ calculus

1. The problem is to find the first derivative of the function $$y = x \sqrt{x} + \frac{3}{x^5}$$ and verify if $$y' = \frac{3}{2} \sqrt{x} - \frac{15}{x^6}$$ is correct. 2. Recall

1. **State the problem:** We have a strip of tin 14 inches long, bent to form a trapezoidal gutter with one side bent up at a 45° angle. We want to find the width of the base that

1. **Problem statement:** We want to set up the integral $$\int_C 6x \, ds$$ where curve $C$ consists of three parts: - Part 1: $y = 3 + x^2$ from $x = -2$ to $x = 0$

1. **Stating the problem:** Buatlah soal tentang konsep dasar integral dan selesaikan.

1. **Énoncé du problème :** Trouver une primitive de la fonction $f(x) = 3x^2(x^3 + 2)^4$ définie sur $\mathbb{R}$.\n\n2. **Formule et règles importantes :** Pour intégrer une fonc

1. **Problem:** Calculate the definite integral $$\int_2^4 (-6x^5 + 3x^4 - 2x^3) \, dx$$. 2. **Formula:** The definite integral of a polynomial function $$f(x)$$ from $$a$$ to $$b$

1. **Problem statement:** Evaluate the integral $$\int_{\pi/6}^{\pi/3} \frac{3\tan^2 x + 6 \tan x + 11}{1 + \tan^2 x} \, dx = \frac{\pi a + b}{6}$$

1. Problem: Find the derivative of the function \( f(x) = \left( 2x^2 - 3x \right) \left( -4x^{-2} + 5 \right) \). 2. Formula: Use the product rule for derivatives: \( (uv)' = u'v

1. **State the problem:** We need to evaluate the double integral over the entire xy-plane:

1. **State the problem:** We need to find the integral of the function $5x$ with respect to $x$. 2. **Recall the formula:** The integral of $x^n$ with respect to $x$ is given by

1. The problem is to evaluate the definite integral $$\int_1^6 x \, dx$$ which represents the area under the curve $y = x$ from $x=1$ to $x=6$. 2. The formula for the integral of $

1. **State the problem:** We want to show that the curvature $\kappa$ of the helix given by the vector function $$\mathbf{r}(t) = \langle a \cos t, a \sin t, b t \rangle$$ is $$\ka

1. **Problem Statement:** Evaluate the integral $$\int_0^\infty x^n e^{-x} \, dx$$ for $n=0,1,2,3$. 2. **Formula and Important Rule:** This integral is known as the Gamma function

1. **State the problem:** We want to show that the curvature $\kappa$ of the helix given by the vector function $$\mathbf{r}(t) = \langle a \cos t, a \sin t, bt \rangle$$ is $$\kap

1. **State the problem:** We need to find the indefinite integral $$\int (1 + x)(1 + 2x) \, dx$$. 2. **Expand the integrand:** Use the distributive property to multiply the two bin

1. **State the problem:** We are given two polynomials:

1. مسئله: باید انتگرال تابع $$3x^2 - 6x + 2\cos x$$ را نسبت به $$x$$ محاسبه کنیم. 2. فرمول‌ها و قواعد مهم: انتگرال جمع برابر است با جمع انتگرال‌ها، یعنی

1. مسئله: باید انتگرال تابع $$3x^2 - 6x + 2\cos x$$ را نسبت به $$x$$ محاسبه کنیم. 2. فرمول‌ها و قواعد مهم: انتگرال تابع جمع برابر است با جمع انتگرال‌ها، یعنی

1. **State the problem:** Find the limit $$\lim_{(x,y) \to (2,2)} \frac{x - y}{x^4 - y^4}.$$\n\n2. **Recall the formula and important rules:** The expression involves a difference

1. **State the problem:** Find the curvature $\kappa$ of the curve given by the vector function $$\mathbf{r}(t) = t\mathbf{i} + t^2\mathbf{j} + t^2\mathbf{k}.$$\n\n2. **Recall the

1. **Problem statement:** Find the derivative of the function $$f(x) = \frac{\sqrt{x-1} - \frac{5}{\sqrt{x-1}}}{\sqrt[3]{x-1}}$$