∫ calculus

1. Let's clarify the question: it seems you are asking why or how the differential operator $d\!u$ might disappear in a mathematical context. 2. The differential $d\!u$ represents

1. The problem is to find the integral $$\int e^x \cdot \sin(e^x) \, dx$$. 2. We use substitution to solve this integral. Let $$u = e^x$$. Then, the derivative is $$\frac{du}{dx} =

1. **State the problem:** We need to evaluate the integral $$\int (8x - 3) \ln x \, dx.$$\n\n2. **Recall the integration by parts formula:** $$\int u \, dv = uv - \int v \, du.$$\n

1. **State the problem:** Find the limit $$\lim_{x \to \pi} 3 + \cos(x)$$. 2. **Recall the limit rule:** If a function is continuous at the point where the limit is taken, then the

1. **State the problem:** Evaluate the definite integral

1. **State the problem:** We need to evaluate the integral $$\int \sqrt[3]{x} \left(4 - \sqrt{x} + \frac{2}{x \sqrt[3]{x}}\right) \, dx.$$\n\n2. **Rewrite the integrand using expon

1. **State the problem:** Find the derivative of the function $$f(x) = 4x^5 - 3x^3 + 2x^2 - 7x + 5$$. 2. **Recall the power rule for derivatives:** For any term of the form $$ax^n$

1. **State the problem:** We need to evaluate the integral $$\int 202x^2 - 4 \, dx$$. 2. **Recall the integral rules:** The integral of a sum is the sum of the integrals, and the i

1. **Stating the problem:** We are given the function $$y(x) = 3\sqrt{x} + 4\cosh x - \frac{2}{x^3} + 5\ln x$$ and we want to understand it clearly step by step. 2. **Understanding

1. **State the problem:** Find the first and second derivatives of the parametric functions $x=t$ and $y=t-\frac{1}{t}$ at $t=4$. 2. **Recall formulas:** For parametric equations $

1. **State the problem:** Evaluate the limits $$\lim_{x \to 7^+} \frac{\sqrt{x-7}}{\sqrt{x^2 - 49}}, \quad \lim_{x \to 7^-} \frac{\sqrt{x-7}}{\sqrt{x^2 - 49}}, \quad \lim_{x \to 7}

1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{6 - 3x + 10x^2}{-2x^4 + 7x^3 + 1}$$ as $x$ approaches 1. 2. **Substitute $x=1$ directly:**

1. **Problem:** Prove that $$\lim_{x \to 1} (-4x + 3) = -1$$ using the definition of limit (\(\varepsilon-\delta\) definition). 2. **Definition:** The limit $$\lim_{x \to a} f(x) =

1. **Problem:** Find the first derivative $f'(x)$ and second derivative $f''(x)$ of the function $f(x) = 3x + 2$. 2. **Formula and rules:**

1. **State the problem:** Find the limit $$\lim_{x \to +\infty} \sqrt[3]{x^2 + x + 1}$$. 2. **Recall the cube root limit rule:** For large $x$, the term with the highest power domi

1. We are asked to find the derivative of the function $$y = \frac{1}{x \sin x}$$. 2. This function is a quotient, so we can rewrite it as $$y = (x \sin x)^{-1}$$ to use the chain

1. **Problem statement:** Find the derivative of the function $f$ for each given case. 2. **Recall the product rule:** For functions $u(x)$ and $v(x)$, the derivative of their prod

1. **State the problem:** Evaluate the double integral $$\int_0^1 \int_x^{\sqrt{x}} y(x + y) \, dy \, dx$$. 2. **Understand the integral:** The integral is over $y$ from $x$ to $\s

1. **State the problem:** We are given two infinite series: $$\sum_{n=1}^{\infty} \frac{(x+2)^n}{n^2} \quad \text{and} \quad \sum_{n=1}^{\infty} \frac{(x+1)^n}{3^n}$$

1. Problem 43: Find the equation of the tangent line to the graph of $y = x^2 \sin 4x$ at $x = \frac{\pi}{4}$. 2. To find the tangent line, we need the point and the slope at $x =

1. **State the problem:** Find the limit $$\lim_{x \to -1} 9 \cdot g(x)$$ where $$g(x) = \begin{cases} 5 - 2x & \text{if } x < -1 \\ 2x & \text{if } x \geq -1 \end{cases}$$. 2. **U