∫ calculus

1. **State the problem:** Evaluate the limit $$\lim_{x \to 0} \frac{e^x - x - 1}{5x^2}.$$\n\n2. **Recall the formula and rules:** When direct substitution results in an indetermina

1. **State the problem:** Evaluate the limit $$\lim_{x \to 1} \frac{\sqrt{x} - \sqrt[3]{x}}{x - 1}$$. 2. **Recognize the indeterminate form:** Substituting $x=1$ gives $$\frac{\sqr

1. The problem is to find the derivative of a geometric function, typically involving shapes or geometric formulas. 2. The derivative measures how a function changes as its input c

1. **State the problem.** Find $\lim_{x\to 0}\dfrac{\sin(3x)-3x}{x^3}.$

1. Vamos resolver o primeiro integral: $$\int x e^{-x^2} \, dx$$ 2. Para resolver, usamos substituição. Seja $$u = -x^2$$, então $$du = -2x \, dx$$ ou $$-\frac{1}{2} du = x \, dx$$

1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{\frac{2}{x} + 3 + x - 1}{x - 5} \div (x - 1).$$ 2. **Rewrite the expression:** The expression can be written as

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$. 2. **Recall the formula and rules:** When direct substitution results in an indeterminate form li

1. **مشكلة:** حساب قيمة ب التي تحقق شروط مبرهنة القيمة المتوسطة للدالة (رس) = س + س جـاس على الفترة [0 ، ب] حيث د ج = \frac{\pi}{2}. 2. **مبرهنة القيمة المتوسطة:** تنص على أنه إذا

1. **State the problem:** We need to evaluate the integral $$\int \frac{27 \sec^3 \theta - 3 \sec \theta \tan \theta}{3 \tan \theta} \, d\theta$$

1. **State the problem:** We want to evaluate the integral $$\int \frac{x^3}{\sqrt{x^2 - 9}} \, dx$$. 2. **Identify the substitution:** Since the integrand contains $\sqrt{x^2 - 9}

1. **Problem statement:** Given the function $f(x) = x^3 - 4x^2 - 4x - 1$, find the first derivative $f'(x)$. 2. **Formula and rules:** The derivative of a function $f(x)$ with res

1. **State the problem:** Find the area under the curve $$y = \frac{x^2 + 3}{4x - x^2}$$ from $$x=1$$ to $$x=3$$. 2. **Formula used:** The area under a curve from $$a$$ to $$b$$ is

1. **Problem statement:** Find the volume of the solid formed by rotating the region under the curve $$y=\frac{5}{x^2+5x+6}$$ from $$x=0$$ to $$x=1$$ about the x-axis. 2. **Formula

1. **State the problem:** Find the equation of the tangent line to the curve $y = f(x) = -2 \sin^2(x)$ at $x = \frac{11\pi}{6}$.\n\n2. **Formula and rules:** The tangent line at $x

1. **State the problem:** Find the equation of the tangent line to the function $f(x) = 3 \sec(x)$ at $x = 0$. 2. **Recall the formula for the tangent line:** The tangent line to $

1. **State the problem:** Evaluate the limit $$\lim_{x \to \infty} \frac{4x^2 + 20x}{5x^2 + 18}$$. 2. **Recall the rule for limits of rational functions:** When $x$ approaches infi

1. **State the problem:** We have a function $h(s)$ representing the height above sea level of a road at position $s$ miles, with domain $[0,10]$. The function has a single critica

1. **Stating the problem:** We are given limits of a function $f(x)$ approaching points $a=2$ and $b=-1$ from the left and right, as well as limits at infinity. We need to find: a)

1. **State the problem:** Differentiate $\sin^3(3x^2 - 5)$ with respect to $x$. 2. **Recall the formula:** If $y = [f(x)]^n$, then by the chain rule, $\frac{dy}{dx} = n[f(x)]^{n-1}

1. **State the problem:** Find the area of Region S bounded by the curves $f(x) = 2 - \frac{1}{4}x^2$ and $g(x) = -\cos\left(\frac{\pi}{3}x\right)$ from $x=0$ to $x=2$.

1. **Problem statement:** Find the area of region S bounded by $f(x) = -\cos\left(\frac{\pi}{3} x\right)$, $g(x) = 2 - \frac{1}{4} x^2$, the y-axis ($x=0$), and the vertical line $