∫ calculus

1. **State the problem:** Find the area of the region bounded by the graph of the function $f(x) = x^3 + 1$, the x-axis, and the vertical lines $x = -2$ and $x = 0$. 2. **Formula a

1. The problem is to evaluate the integral $$I = \int \tan(\tan t) \cdot t \, dt$$. 2. This integral involves a composition of trigonometric functions and a polynomial term, which

1. The problem is to evaluate the integral $$I = \int \tan(\tan t) \cdot t \, dt.$$\n\n2. This integral is quite complex because it involves a composition of the tangent function i

1. **Problem:** Find the derivative $\frac{dy}{dx}$ for $y = x \cos 2x$ using the product rule. 2. **Formula:** The product rule states that if $y = u(x)v(x)$, then

1. **Problem Statement:** Determine if the following improper integrals converge or diverge: (i) $$\int_4^{\infty} 8^x \, dx$$

1. Problem statement: Find the limit as $x$ approaches $-5$ of the function. $$\lim_{x\to -5} \frac{\frac{1}{5}+\frac{1}{x}}{10+2x}$$

1. Problem: Compute the limit as $x \to -5$ of $\frac{\frac{1}{5} + \frac{1}{x}}{10} + 2x$. 2. Formula and rules: We use the limit laws: $\lim_{x\to a}(f(x)+g(x))=\lim_{x\to a}f(x)

1. **State the problem:** Find the limit as $x$ approaches $-5$ of the expression $$\frac{\frac{1}{5} + \frac{1}{x}}{10 + 2x}.$$\n\n2. **Rewrite the expression:** The expression ca

1. **State the problem:** Find the limit as $x$ approaches $-5$ of the expression $$\frac{\frac{1}{5} + \frac{1}{x}}{10} + 2x.$$\n\n2. **Rewrite the expression:** The expression ca

1. **State the problem:** Find the limit as $x$ approaches $-5$ of the expression $$\frac{\frac{1}{5} + \frac{1}{x}}{10} + 2x.$$\n\n2. **Rewrite the expression:** The expression ca

1. **State the problem:** We need to find the limit as $x$ approaches 5 of the expression $$\frac{\frac{1}{5+x}}{10+2x}.$$\n\n2. **Rewrite the expression:** The expression can be s

1. **State the problem:** Find the limit $$\lim_{x \to 5^+} \frac{\frac{1}{5} + \frac{1}{x}}{10 + 2x}$$. 2. **Recall the limit properties:** If the function is continuous at the po

1. The problem is to find the derivative $f'(x)$ of the function represented by the blue line graph described. 2. The graph starts near $(-5,4)$, goes down to a minimum near $(-2,-

1. **Problem statement:** Find the equation of the tangent line to the curve $y=\cos x$ at $x=1$. 2. **Recall the formula for the tangent line:**

1. **Problem statement:** Find the equation of the tangent line to the curve $y=\cos x$ at the point where $x=1$. 2. **Formula used:** The equation of the tangent line to a functio

1. Problem 7: Evaluate $$\iint_D y^2 \, dA$$ where $$D = \{(x,y) \mid -1 \le y \le 1, -y - 2 \le x \le y\}$$. 2. The integral is set up as an iterated integral with respect to $$x$

1. **Problem statement:** Evaluate the double integral $$\iint_D y^2 \, dA$$ where $$D = \{(x,y) \mid -1 \leq y \leq 1, -y - 2 \leq x \leq y\}$$. 2. **Set up the integral:** The re

1. **State the problem:** Evaluate the double integral $$\int_0^2 \int x y^x \, dy \, dx$$. 2. **Understand the integral:** The integral is over $y$ first, then $x$. The inner inte

1. The problem is to find the radius and interval of convergence of the series $$\sum_{n=0}^\infty (x+5)^n$$ and determine for which values of $x$ the series converges. 2. This is

1. The problem is to find the derivatives of basic trigonometric functions such as $\sin x$, $\cos x$, and $\tan x$. 2. The fundamental formulas for derivatives of trigonometric fu

1. **State the problem:** We need to find the derivative $\frac{dz}{dx}$ when $z = x e^{xy}$, where $y$ is treated as a constant with respect to $x$. 2. **Recall the formula:** To