∫ calculus

1. **State the problem:** Find the derivative of the function $f(x) = \frac{1}{x^2+1}$ using first principles (the definition of the derivative). 2. **Recall the definition of the

1. **State the problem:** Find the limits:

1. **State the problem:** Find the derivative of the function $f(x) = \frac{1}{x^2} + 1$ using first principles (the definition of the derivative). 2. **Recall the definition of th

1. **Stating the problem:** We want to find the limit $$\lim_{x \to 0} \frac{4x + \sin 3x}{6x - \tan 4x}$$

1. **Problem statement:** We analyze the extrema of the functions \(f_a(x) = x^2 - ax - 2\), \(g_a(x) = 2ax^3 - 6x\), and \(h_a(x) = -\frac{1}{5}x^3 + \frac{3}{5}ax^2\) depending o

1. **State the problem:** Evaluate the limit $$\lim_{x \to 4} \frac{x^2 - 16}{x - 4}$$. 2. **Recall the formula and rules:** When direct substitution in a limit results in an indet

1. **Problem Statement:** Evaluate the integral $$\int_0^{\pi/2} \frac{dx}{4\cos x + 2\sin x}.$$\n\n2. **Formula and Approach:** To solve integrals of the form $$\int \frac{dx}{a\c

1. **Problem Statement:** Find the intervals where the function $f(x) = (x+1)^3 (x-3)^3$ is strictly increasing or strictly decreasing. 2. **Formula and Rules:** To determine incre

1. The problem is to find the limit $$\lim_{x\to 0}\frac{\arcsin(x)}{x}$$. 2. Recall the important limit rule: $$\lim_{x\to 0}\frac{\sin x}{x} = 1$$ and the fact that $$\arcsin(x)$

1. **State the problem:** Find the limit $$\lim_{x \to \pi} \frac{\sin x}{\pi - x}$$. 2. **Recall the formula and important rule:** This is a limit of the form $$\frac{\sin x}{\pi

1. مسئله را بیان می‌کنیم: می‌خواهیم مقدار

1. مسئله را بیان می‌کنیم: مقدار $a$ را در معادله $$\frac{4}{3} = \lim_{x \to \frac{1}{2}^+} a \left(-\frac{2}{x^3}\right) + 2x \over 4x - \frac{1}{x}$$

1. The problem: Understand what an integral is and how it is used in calculus. 2. Definition: An integral is a mathematical tool used to find the area under a curve or to accumulat

1. Mari kita tentukan titik ekstrim dan titik belok fungsi kubik. 2. Fungsi kubik umum: $$y = ax^3 + bx^2 + cx + d$$

1. **State the problem:** We are given the function $s = \sqrt{t} + 2$ and need to find the derivative $\frac{ds}{dt}$ at $t = 7$. 2. **Recall the formula:** The derivative of $s$

1. Mari kita selesaikan soal pertama bagian a: Derivasi fungsi $Y = (2X + 5)(4X^2)$. 2. Gunakan aturan perkalian untuk turunan: Jika $Y = u \cdot v$, maka $Y' = u'v + uv'$.

1. **Problem 1: Differentiate $y = x \sin x$ with respect to $x$.** 2. Use the product rule: If $y = uv$, then $\frac{dy}{dx} = u'v + uv'$. Here, $u = x$, $v = \sin x$.

1. **Problem:** Evaluate the integral $$\int 3x^5 \sqrt{16 - x^2} \, dx$$. 2. **Formula and rules:** We will use substitution and algebraic manipulation. Recall that substitution i

1. **Problem 1a:** Find the derivative of $y = 5x^5$. Formula: For $y = ax^n$, the derivative is $y' = a n x^{n-1}$.

1a. Differentiate $y=5x^5$ using the power rule $\frac{d}{dx}x^n = nx^{n-1}$. $$\frac{dy}{dx} = 5 \times 5x^{5-1} = 25x^4$$

1. The problem is to evaluate the improper integral $$\int_0^\infty \frac{x}{(1+x^2)^2} \, dx$$ over the infinite interval from 0 to infinity. 2. We use the formula for integration