∫ calculus

1. The problem asks us to understand the definition and properties of continuity of a function at a point $a$. 2. A function $f$ is continuous at $a$ if three conditions are met:

1. **Problem Statement:** Find the value of the integral $$\int_0^\infty \frac{1}{1+x^n} \, dx$$ for $$n > 1$$. 2. **Formula and Important Rules:** This integral is a known form re

1. **State the problem:** We need to find the integral of $\cos 2x$ with respect to $x$. 2. **Recall the formula:** The integral of $\cos(ax)$ is $\frac{1}{a} \sin(ax) + C$, where

1. **State the problem:** We need to find the integral of the function $x \cos(2x)$ with respect to $x$. 2. **Formula and method:** We will use integration by parts, which states:

1. **State the problem:** We need to find the integral of the function $2 \sin(2x)$. 2. **Recall the formula:** The integral of $\sin(ax)$ with respect to $x$ is $-\frac{1}{a} \cos

1. The problem states that the derivative of a function $A(x)$, denoted as $A'(x)$, is 24. 2. This means the rate of change of $A(x)$ with respect to $x$ is constant and equal to 2

1. Problem: Find $\frac{d^2y}{dx^2}$ for the function $y = 5x^2 - 7x + 3$. 2. Formula: Use the power rule $\frac{d}{dx}x^n = n x^{n-1}$ and the linearity of differentiation which a

1. The problem asks us to understand the meaning of the limit statement: $$\lim_{x \to 2} f(x) = 6$$ and whether it is still true if $$f(2) = 7$$. 2. The limit $$\lim_{x \to a} f(x

1. **Problem 1:** Find the derivative $f'(x)$ if $f(x) = (x^3 - x)(x^2 - 2)$. - Use the product rule: $\frac{d}{dx}[u \cdot v] = u'v + uv'$.

1. **Problem 1: Find the equation of the tangent line to the graph of** $f(x) = x^2\sqrt{2x + 12}$ **at** $x=2$. 2. **Formula:** The tangent line at $x=a$ is given by

1. The problem asks to evaluate a limit, but since no image was provided, I cannot see the exact limit expression. 2. To solve a limit problem, we typically use substitution, facto

1. **State the problem:** Given the function $f(x) = \frac{1}{3}x^3 - \frac{1}{2}x^2 + 6x$, find the values of $a$ such that $f'(a) = 18$. 2. **Find the derivative:** Use the power

1. The problem asks: What is the limit of the function as $x$ approaches 0 based on the given graph description? 2. From the description, the function has a vertical asymptote at $

1. The problem asks us to estimate the limit $$\lim_{x \to 4} g(x)$$ using the given table of values for the function $g$ near $x=4$. 2. The limit $$\lim_{x \to a} f(x)$$ is the va

1. The problem asks us to estimate the limit $$\lim_{x \to 6} f(x)$$ given the table of values for $f(x)$ near $x=6$. 2. The limit $$\lim_{x \to a} f(x)$$ is the value that $f(x)$

1. **State the problem:** We need to find the indefinite integral $$\int (x^2 - 5)^3 \, dx$$. 2. **Formula and approach:** To integrate a composite function raised to a power, we c

1. **State the problem:** We need to find the indefinite integral $$\int (x^2 - 5)^3 \, dx$$. 2. **Formula and approach:** To integrate a composite function raised to a power, we c

1. The problem is to evaluate the integral $$\int (x^2 - 5)^3 \, dx$$. 2. We use the substitution method for integration. Let $$u = x^2 - 5$$. Then, the derivative is $$\frac{du}{d

1. **State the problem:** Evaluate the integral $$\int (x^2 - 5)^3 \, dx$$. 2. **Formula and substitution:** Use substitution for integrals of composite functions. Let $$u = x^2 -

1. **State the problem:** Calculate the value of

1. **State the problem:** Calculate the value of