∫ calculus

1. **Problem:** Find the derivative of \(y = (3x^2 + 1)^{3/2}\). 2. **Formula:** Use the chain rule for derivatives: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\).

1. **State the problem:** Evaluate the integral $$\int \frac{dx}{\sqrt{8x - x^2}}$$. 2. **Rewrite the expression inside the square root:**

1. **State the problem:** We need to find the derivative of the function $$f(x) = \sqrt{1 + \sqrt{1 + x}}$$ and then evaluate it at $$x=8$$. 2. **Recall the formula:** The derivati

1. **State the problem:** We want to evaluate the integral $$\int \frac{x^3 + 2x}{x^2 - 5x + 6} \, dx$$ using partial fractions. 2. **Factor the denominator:** The denominator fact

1. **Problem:** Express the limit of the sum $$S_n = \lim_{n \to \infty} \sum_{i=1}^n \frac{3i + 2i}{n^r}$$ as an integral. 2. **Formula and Explanation:** The sum $$\sum_{i=1}^n f

1. **Problem statement:** Evaluate the Taylor series for the function $f(x) = \cos x$ centered at $x=0$. 2. **Formula:** The Taylor series of a function $f(x)$ at $x=a$ is given by

1. **State the problem:** We want to find the Taylor series expansion of the function $f(x) = \cos x$ centered at $x=0$. 2. **Recall the Taylor series formula:** The Taylor series

1. **Problem statement:** Find the derivative $\frac{dy}{dx}$ of the function $y = f(x) = \sqrt[3]{x^2}$. 2. **Rewrite the function:** Recall that $\sqrt[3]{x^2} = x^{\frac{2}{3}}$

1. **State the problem:** Evaluate the integral $$\int \frac{\left(\frac{5}{3}\right)^5 \sec^5 \theta}{9 \cdot \left(\frac{5}{3}\right)^2 \sec^2 \theta - 25}^{3/2} \cdot \frac{5}{3

1. **Evaluate the integral** $$\int_0^1 \int_1^2 x(x + y) \, dy \, dx$$. 2. **Determine the integral** $$\int_{-1}^2 (x^3 - 2x) \, dx$$.

1. **Problem Statement:** Find the area bounded by the parabola $y^2 = 4ax$ and the parabola $x^2 = 4ay$ using double integration. 2. **Understanding the curves:**

1. **Problem:** Evaluate the integral $$\int \ln(3x - 2) \, dx$$ 2. **Formula and rules:** Use integration by parts formula:

1. Problem: Find the differential coefficient (derivative) of $\tan(\sin(ax+b))$. 2. Formula and rules: Use the chain rule for derivatives. If $y = f(g(x))$, then $\frac{dy}{dx} =

1. **הבעיה:** נתונה הפונקציה $$f(x) = \frac{3 - x}{(x - 2)^2}$$ יש לפתור את כל הסעיפים א-ז ולענות על השאלות הנוספות. 2. **תחום ההגדרה:** הפונקציה מוגדרת לכל ערך של $$x$$ פרט לנקודו

1. נתחיל בפתרון סעיף א: תחום ההגדרה של הפונקציה הפונקציה נתונה כ-$$f(x) = \frac{3 - x}{(x - 2)^2}$$

1. The problem is to find the normalizing constant $c$ such that $$\int_{-\infty}^{\infty} c e^{-x^2/2} \, dx = 1.$$\n\n2. We use the fact that the integral of the Gaussian functio

1. The problem is to solve the integral using improper integrals. 2. Improper integrals are used when the interval of integration is infinite or the integrand has an infinite disco

1. **Problem Statement:** Given the equation $$y^2 + 2ax - a^2 = 0,$$ show that $$\left(\frac{dy}{dx}\right)^2 + \frac{2x}{y} \frac{dy}{dx} = 1.$$\n\n2. **Step 1: Differentiate the

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the ellipse defined by the equation $$25x^2 + y^2 - 16 = 0$$ at the point $x = -1$.

1. Masalah: Hitung integral lipat dua $$\int_0^1 \int_0^{\sqrt{1-y^2}} \sin(x^2 + y^2) \, dx \, dy$$ 2. Formula dan aturan: Integral lipat dua digunakan untuk menghitung volume di

1. Problem: Calculate the integral $$\int x^2 (8x^3 - 6) \, dx$$ 2. Use the distributive property to expand the integrand: