∫ calculus

1. **Problem:** Do an example of an integral. 2. **Formula used:** For a basic power integral, use

1. **Stating the problem:** Find the derivative of the function $$f(x) = 2x^3 - 9x^2 + 12x$$ and then evaluate it at $$x = 2$$ to find $$f'(2)$$.

1. **Stating the problem:** We are given the equation $x^5 = 2x^3 + c$ and a point $(3,5)$ on the curve. We need to find the constant $c$ and verify the gradient (derivative) at $x

1. **Problem 3.1:** Find the points of intersection in the first quadrant of the polar curves $r_1 = \sin \theta$ and $r_2 = \sin(2\theta)$.\n\n2. The points of intersection satisf

1. **State the problem:** We need to find the indefinite integral $$\int 18x^2 (x^3+2)^5 \, dx.$$\n\n2. **Identify the method:** This integral suggests using substitution because t

1. The problem is to find the extrema of the function $$y = (x - 2)^4 - 1$$ using higher order derivatives. 2. To find extrema, we first find the first derivative $$y'$$ and set it

1. We are asked to evaluate the definite integral $$\int_0^1 \frac{x^2}{\sqrt{x^3 + 1}} \, dx$$. 2. To solve this integral, we use substitution. Let $$u = x^3 + 1$$. Then, $$\frac{

1. We are asked to evaluate the integral $$\int \cos^7\left(\frac{2x}{5}\right) \cdot \sin^6\left(\frac{2x}{5}\right) \, dx.$$\n\n2. The integral involves powers of sine and cosine

1. **State the problem:** Find the length of the curve given by the vector function $$\mathbf{r}(t) = (\sqrt{2}t, e^t, e^{-t})$$ for $$0 \leq t \leq 1$$. 2. **Formula for curve len

1. **State the problem:** We are given the vector function $$r(t) = \left(\tan(t) \cot(t), \ln t, \sqrt{\pi^2 - t^2}\right)$$ and need to find its domain and the limit $$\lim_{t \t

1. **State the problem:** Find the area bounded by the curves $y = x^2$ and $y = x$. 2. **Identify the points of intersection:** Set $x^2 = x$ to find where the curves intersect.

1. **State the problem:** Evaluate the integral $$\int \frac{x^4}{\sqrt{4 - x^2}} \, dx.$$\n\n2. **Recall the formula and substitution:** For integrals involving expressions like $

1. **Problem:** Evaluate the integral $$\int \frac{dx}{\sqrt{x} + \sqrt[3]{x}}.$$\n\n2. **Formula and rules:** To integrate expressions involving roots, rewrite roots as fractional

1. **Problem statement:** Given the function $g(x) = (x + 1)e^{ax} + b$ with $a,b \neq 0$, and the curve $(G)$ of $g$ has an inflection point at $x=1$ and a horizontal asymptote $y

1. The problem is to evaluate the definite integral $$\int_{e^2}^{e^3} \frac{1}{x} \, dx$$. 2. Recall the integral formula for $$\int \frac{1}{x} \, dx = \ln|x| + C$$, where $$\ln$

1. نبدأ ببيان المشكلة: لدينا دالة ت(س) معرفة على شكل دالة قطعة (piecewise) كالتالي: $$

1. **Problem statement:** Calculate the area of the region bounded by the curve of the inverse function $f^{-1}$, the line $y=x$, and the vertical lines $x=\alpha$ and $x=0$ by ded

1. **Problem:** Find the derivative of $$y = \frac{x^3}{3} + \frac{x^2}{2} + \frac{x}{4}$$ Formula: Use the power rule $$\frac{d}{dx} x^n = n x^{n-1}$$

1. The problem is to find the derivative $\frac{dy}{dx}$ given the implicit equation $$xy + y^2 = 4.$$ 2. We use implicit differentiation, differentiating both sides with respect t

1. **State the problem:** We need to find the integral $$\int e^{2x} \cos(3x) \, dx$$. 2. **Formula and method:** This is an integral of the form $$\int e^{ax} \cos(bx) \, dx$$. Th

1. Diberikan limit $$\lim_{x \to 2} \frac{\sqrt{ax + 2b} + 8}{x - 2} = 7$$. Kita diminta mencari nilai $$a + b$$. 2. Karena limit ini berbentuk $$\frac{f(x)}{x-2}$$ dan limitnya ad