∫ calculus

1. **State the problem:** Evaluate the integral $$\int \cos 4\theta \sqrt{2} - \sin 4\theta \, d\theta$$. 2. **Rewrite the integral:** Separate the integral into two parts:

1. The problem is to find the derivative of the inverse function of a given function. 2. The formula for the derivative of the inverse function $f^{-1}$ at a point $y$ is:

1. **بيان المسألة:** لدينا الدالة $f$ معرفة كما يلي:

1. **State the problem:** We need to find the limit of the function $f(x)$ as $x$ approaches $-3$, i.e., $\lim_{x \to -3} f(x)$. 2. **Recall the definition of a limit:** The limit

1. **Stating the problem:** We need to find the integral $$\int (x^2 + 4x + 7) \sin^{-1}(2x - 1) \, dx$$. 2. **Formula and approach:** This is an integral involving a product of a

1. **Statement of the problem:** We have a piecewise function defined as: $$f(x) = \begin{cases} e^x - 1 & x > 0 \\ \ln(x + 1) & x = 0 \\ x^3 e^{2x} & x < 0 \end{cases}$$

1. **State the problem:** We have the curve $$y = x^3 - 4x^{5/2} - kx^{1/2} + 28x - 44$$ for $$x \geq 0$$, where $$k$$ is a positive constant.

1. **Statement of the problem:** Calculate the derivatives of the following functions:

1. **State the problem:** Find the limit of the piecewise function $$f(x) = \begin{cases} x^2 & \text{if } x \leq 2 \\ 3x - 3 & \text{if } x > 2 \end{cases}$$

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{e^{2x}}{x^{1/2}(x^2+5)^{1/4}}$$. 2. **Rewrite the function for clarity:**

1. **Stating the problem:** We want to analyze and graph the function $$f(x) = 2x + \ln(x^2 - 3)$$ with the given domain and properties.

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{e^{2x}}{x^{1/2}(x^2+5)^{1/4}}$$. 2. **Rewrite the function:** To differentiate, rewrite the function as

1. **State the problem:** Find the limit as $x \to +\infty$ of the expression $$\frac{\sqrt{x} + \sqrt{x} + \sqrt{x}}{\sqrt{x} + 1}.$$\n\n2. **Simplify the numerator:** The numerat

1. **State the problem:** Find the limit $$\lim_{x \to +\infty} \frac{\sqrt{x} + \sqrt{x + \sqrt{x}}}{\sqrt{x} + 1}$$ as $x$ approaches positive infinity. 2. **Recall the formula a

1. Problem: Find the first derivative $\frac{dy}{dx}$ for $y=2^x \sin^{-1} x$. Use the product rule since $y$ is a product of two functions. 2. Formula: Product rule states $\frac{

1. **State the problem:** Find the limits using the definition of the derivative (first principle): $$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

1. **Stating the problem:** We want to analyze and graph the function $$f(x) = \frac{x^2}{(x-1)^2}$$ with the given properties about domain, monotonicity, local minima, concavity,

1. **State the problem:** We need to evaluate the integral $$\int \left( \frac{3}{\sqrt{1 - x^2}} + \frac{4}{\sqrt{1 + x^2}} + \frac{5x}{\sqrt{x^2 - 1}} \right) dx$$

1. **Problem Statement:** Given the function $f(x) = e^x$, show that its derivative $f'(x) = e^x$. 2. **Formula Used:** The derivative of the exponential function $e^x$ with respec

1. We are asked to evaluate the integral $$2 \int \frac{\sin x}{3 + \cos 2x} \, dx$$. 2. Recall the double-angle identity: $$\cos 2x = 2\cos^2 x - 1$$.

1. **Stating the problem:** We need to evaluate the expression