∫ calculus

1. مسئله: انتگرال $$I_n = \int \frac{1}{x^n \sqrt{ax+b}} \, dx$$ را به صورت رابطه بازگشتی بیابید. 2. ابتدا قاعده انتگرال‌گیری جز به جز را یادآوری می‌کنیم: $$\int u \, dv = uv - \in

1. مسئله: انتگرال $$I_n = \int \frac{1}{x^n \sqrt{ax+b}} \, dx$$ را به صورت رابطه بازگشتی بیابیم. 2. ابتدا تعریف کنیم: $$I_n = \int \frac{1}{x^n \sqrt{ax+b}} \, dx$$ که در آن $$n \

1. Bài toán yêu cầu tính đạo hàm bậc 100 của hàm số $$f(x) = \frac{1}{x^2 + 4}$$. 2. Ta nhận thấy hàm số có dạng phân thức với mẫu là đa thức bậc 2.

1. **Problem:** Calculate the definite integral $$\int_0^3 x \, dx$$ 2. **Formula:** The integral of $$x$$ is $$\frac{x^2}{2}$$. For definite integrals, evaluate the antiderivative

1. Given the problem: Find the differential $dz$ for the function $$z = x^3 + x^2 y - x y^2 + 4 y^3.$$ 2. The formula for the total differential of a function $z = f(x,y)$ is:

1. **Problem:** Given the function $f(q) = hq^2 + mq + c$ with gradient function $4q + 8$ and a stationary value of $-3$, find $h$, $m$, and $c$. 2. **Step 1: Understand the gradie

1. **Problem:** Find $\frac{dy}{dx}$ using implicit differentiation for the equation $x^2 + y^2 = 25$. 2. **Formula:** Use the rule $\frac{d}{dx}[y^n] = n y^{n-1} \frac{dy}{dx}$ wh

1. **Problem Statement:** We need to sketch a function $f(x)$ defined on $\mathbb{R}$ that satisfies the following limit conditions:

1. **State the problem:** Evaluate the limit as $r$ approaches 0 of the expression $$\frac{5r^{-0.4} - 7r^{-\frac{3}{5}} + 3r^{-0.1} - 4r^{-\frac{3}{7}} - 8r^{-0.2}}{}$$

1. **State the problem:** Evaluate the limit as $r$ approaches 0 of the expression $$\frac{5r^{-0.4} - 7r^{-\frac{3}{5}} + 3r^{-0.1} - 4r^{-\frac{3}{7}} - 8r^{-0.2}}{}$$

1. We are asked to find the limit: $$\lim_{x \to \left(\frac{1}{2}\right)^-} \frac{\ln(1 - 2x)}{\tan(\pi x)}.$$\n\n2. First, note the behavior of the function inside the logarithm

1. **Problem Statement:** Find the value of $x$ where the slope of the tangent to the curve $y = x^2 + 3x + 2$ is equal to 7. 2. **Formula and Rules:** The slope of the tangent to

1. The problem is to find the value of $x$ where the slope of the tangent to a curve is given or needs to be determined. 2. The slope of the tangent line to a curve at a point is g

1. The problem is to find the critical points of a function. 2. Critical points occur where the derivative of the function is zero or undefined.

1. **State the problem:** We want to analyze the function $y = -x^3 + 6x^2$ and understand its graph using the slope of the tangent line. 2. **Formula for slope of tangent line:**

1. The problem is to find the sixth order derivative of the function $y = \arctan(x)$. 2. Recall that the first derivative of $\arctan(x)$ is given by the formula:

1. Let's first state the problem: Determine if $x=0$ is a critical point of a given function. 2. Recall the definition: A critical point of a function $f(x)$ occurs where the deriv

1. **State the problem:** We want to find the intervals on which the function $$f(x) = \cos^2(4x) + 3 \cos(4x)$$ is increasing or decreasing for $$0 < x < \frac{\pi}{2}$$. 2. **Fin

1. **State the problem:** We need to find the derivative of the composite function $ (f \circ g)(x) = f(g(x)) $ at $ x=2 $. Given functions are $ f(x) = 3x^{2} - 2x + 1 $ and $ g(x

1. **State the problem:** Evaluate the limit $$\lim_{x \to 0} \frac{3x}{4 - 4\sec^2(5x)}.$$\n\n2. **Recall the formula and important rules:** The secant function is $$\sec(\theta)

1. **Problem statement:** We need to find the instantaneous velocity of an object moving in a straight line at time $t=3$ seconds, given its position function $$s(t) = 8\sqrt{t+1}