∫ calculus

1. **Problem Statement:** Evaluate the integral $$\int_0^{10} g(x) \, dx$$ by interpreting it in terms of areas under the graph of $$g(x)$$. 2. **Understanding the graph:** From $$

1. **State the problem:** Evaluate the indefinite integral $$\int x \sqrt{8 - x^2} \, dx$$. 2. **Recall the formula and substitution method:** When integrating expressions involvin

1. **State the problem:** Evaluate the indefinite integral $$\int x^2 \sqrt{x^3 + 7} \, dx$$ using the substitution $$u = x^3 + 7$$. 2. **Identify the substitution and its derivati

1. **Problem:** Evaluate $$\lim_{x \to -1} \frac{x^2 - x - 2}{x^3 - 6x^2 - 7x}$$ 2. **Formula and rules:** When direct substitution leads to an indeterminate form like $$\frac{0}{0

1. **Problem Statement:** Find the derivative of a composite function using the chain rule. 2. **Formula:** The chain rule states that if you have a composite function $y = f(g(x))

1. The problem is to find the integral of $\ln(2x)$ with respect to $x$. 2. We use the integration formula for logarithmic functions: $$\int \ln(ax)\,dx = x\ln(ax) - x + C$$ where

1. The problem is to understand the sum and difference rules for limits in calculus. 2. The sum rule states that the limit of a sum is the sum of the limits: $$\lim_{x \to a} [f(x)

1. **State the problem:** We need to find the third derivative of the function $$t=\sin(t^{2}+8)+4e^{3}x-\cos(e^{x})$$ with respect to $x$. 2. **Rewrite the function:** Note that $

1. The problem asks for the third and fourth terms in the Taylor series expansion of a function $f(x)$ about the point $a$. 2. The Taylor series formula for a function $f(x)$ expan

1. The problem asks for the third term in the Taylor series expansion of a function $f(x)$ around the point $a$. 2. The Taylor series expansion of $f(x)$ about $a$ is given by:

1. **Problem Statement:** We are given a piecewise function: $$f(x) = \begin{cases} x^2 \cos\left(\frac{\pi}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}$$

1. **Problem:** Find the derivative of the function $y = 2 \ln x$. 2. **Formula:** The derivative of $\ln x$ with respect to $x$ is $\frac{1}{x}$.

1. **Problem:** Find the derivative of the function $y = 2 \ln x$. 2. **Formula and rules:** The derivative of $\ln x$ with respect to $x$ is $\frac{1}{x}$ for $x > 0$.

1. **Problem:** Express the limit \(\lim_{\|P\|\to 0} \sum_{k=1}^n c_k^2 \Delta x_k\) as a definite integral, where \(P\) is a partition of \([0, 2]\). 2. **Formula and Explanation

1. State the problem: Use the Constant Multiple Rule in limits, which states that $$\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)$$ where $c$ is a constant. 2. Exampl

1. **Problem:** Find the value of $c$ that satisfies Rolle's Theorem for $$f(x) = \frac{x^2 + 4x - 12}{x^2 + 2x - 3}$$

1. **State the problem:** We are given the function $f(x) = x^2 + 1$ and asked to analyze it using calculus. 2. **Recall the formula for the derivative:** The derivative of a funct

1. **Problem:** Find the value of $c$ that satisfies Rolle's Theorem for $$f(x) = \frac{x^2 + 4x - 12}{x^2 + 2x - 3}$$ on the interval $[-6, 2]$.

1. The problem is to find the derivative of the function $f(x) = 3x^3 - 2$ using calculus. 2. The formula for the derivative of a power function $x^n$ is given by the power rule: $

1. The problem is to analyze the function $f(x) = x^2 - 4$ using calculus. 2. We start by finding the critical points where the derivative is zero or undefined. The derivative of $

1. The problem is to understand and apply the theorem on limits: $$\lim_{x \to a} c = c$$ where $c$ is a constant and $a$ is the point $x$ approaches. 2. This theorem states that t