∫ calculus

1. **State the problem:** Evaluate the double integral $$\int_0^1 \int_{\frac{y}{2}}^2 (1 - y^2) \, dx \, dy$$. 2. **Understand the integral:** The integrand is $$1 - y^2$$, which

1. **Problem Statement:** Evaluate the integral $$\int (2x^3 + 1)^7 x^2 \, dx$$ using the change of variable method. 2. **Recall the formula:** For integrals of the form $$\int (g(

1. **State the problem:** We are given the polynomial function $$f(x) = \frac{15}{4}x^4 + 10x^3 - 60x^2 + 30$$ and need to find its local extrema (local minima and maxima). 2. **Fo

1. **State the problem:** Evaluate the integral $$\int (4x^2 - 8x + 1) \, dx$$. 2. **Recall the integral rules:**

1. Stating the problem: We analyze the differentiability and continuity of the functions graphed in Exercises 43-48 over their given domains. 2. Important concepts:

1. **State the problem:** We need to evaluate the integral $$\int \frac{x^2}{\sqrt{2x+1}} \, dx$$. 2. **Use substitution:** Let $$u = 2x + 1$$, then $$du = 2 \, dx$$ or $$dx = \fra

1. **State the problem:** We want to find the limit as $x$ approaches 1 of the expression $$\sqrt[3]{\frac{(3x^2 + 3x - 1)^2}{5x^3 (x^2 - 1)}}.$$\n\n2. **Rewrite the expression:**

1. **State the problem:** We want to find constants $A$ and $B$ such that $$\frac{x}{(x-4)(x-1)} = \frac{A}{x-4} + \frac{B}{x-1}$$

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$. 2. **Recall the formula and rules:** The expression is a rational function that becomes indetermi

1. Problemi: Të gjejmë integralin $$\int \frac{3}{(x-7)^5} \, dx$$. 2. Formula dhe rregullat: Përdorim formulën për integralin e fuqisë së funksionit $$\int x^n \, dx = \frac{x^{n+

1. Problemi kërkon përdorimin e shenjave të integralit për të zgjidhur një problem të dhënë. 2. Shenjat e integralit përdoren për të llogaritur zonën nën një kurbë ose për të gjetu

1. Problemi: Të gjendet integrali $$\int \frac{3}{(x-7)^5} \, dx$$. 2. Formula dhe rregullat: Përdorim formulën për integralin e fuqisë së funksionit $$\int x^n \, dx = \frac{x^{n+

1. Problemi: Të gjejmë integralin e $$\int \frac{3}{(x-7)^5} \, dx$$. 2. Formula dhe rregullat: Përdorim formulën për integralin e fuqisë së funksionit $$\int x^n \, dx = \frac{x^{

1. نبدأ بكتابة المشكلة: لدينا دالة كثيرة حدود معرفة بقاعدتين: $$h(x) = \begin{cases} \frac{m(x-2)}{x-3}, & x \neq 3 \\ 2x^2 - 11, & x=3 \end{cases}$$

1. **Problem statement:** Find the derivative \( \frac{d}{dx}(\ln \ln 2x) \) and then find \( \frac{dy}{dx} \) for \( y = \ln \ln \ln 2x \).

1. **State the problem:** We are given the integral equation $$\int_0^{x^2} f(t) \, dt = x^2(1 + x)$$ and asked to find the value of $$f(4)$$. 2. **Recall the Fundamental Theorem o

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$. 2. **Recall the formula and rules:** This is a limit of a rational function where direct substitu

1. **State the problem:** We want to find the indefinite integral

1. **State the problem:** Evaluate the definite integral $$\int_{-1}^{\alpha} e^{-x} \, dx$$ and write the result to four decimal places. 2. **Recall the formula:** The integral of

1. **State the problem:** Evaluate the definite integral $$\int_1^9 \frac{x - 1}{\sqrt{x}} \, dx$$ which is rewritten as $$\int_1^9 \left(x^{\frac{1}{2}} - x^{-\frac{1}{2}}\right)

1. **State the problem:** Find the limit as $x$ approaches infinity of the function $4x^2 - 7$. 2. **Recall the limit rule for polynomials:** For a polynomial $ax^n + \dots$, as $x