∫ calculus

1. **State the problem:** We need to find the integral of the function $x^2 e^{2x}$ with respect to $x$. 2. **Formula and method:** We will use integration by parts, which states:

1. **Problem Statement:** Differentiate the function $\sin^2 u$, which means find $\frac{d}{du}(\sin^2 u)$. 2. **Formula Used:** Use the chain rule for differentiation: if $y = [f(

1. The problem is to evaluate the expression $e^{\infty^2}$. 2. Recall that $\infty$ represents an unbounded quantity that grows without limit.

1. **State the problem:** Determine the value of $k$ such that the piecewise function $$f(x) = \begin{cases} kx + 3, & x < 1 \\ 8 - x^2, & x \geq 1 \end{cases}$$

1. **Problem Q.4 (i):** Use the Sandwich Theorem to find $$\lim_{x \to 0} f(x)$$ given $$\sqrt{5 - 2x^2} \leq f(x) \leq \sqrt{5 - x^2}$$. 2. The Sandwich Theorem states if $$g(x) \

1. **State the problem:** We are given a function $g(x) = x f(x)$, with $f(3) = 4$ and $f'(3) = -2$. We need to find the equation of the tangent line to the graph of $g$ at $x = 3$

1. **State the problem:** Find the equation of the tangent line to the curve $y = x\sqrt{x}$ that is parallel to the line $y = 1 + 3x$. 2. **Identify the slope of the given line:**

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: $$y = ax^3 + bx^2 + cx + d$$