∫ calculus

1. **State the problem:** Find the second derivative of the function $f(x) = x^3 + 5x$. 2. **Recall the first derivative:** From the previous problem, the first derivative is $f'(x

1. **State the problem:** Find the derivative of the function $f(x) = x^3 + 5x$. 2. **Recall the derivative rules:**

1. **Problem Statement:** Explain in simple words what an asymptote means. 2. **Definition:** An asymptote is a line that a graph of a function gets closer and closer to but never

1. **Problem Statement:** Find the asymptotes of the function $f(x) = \frac{x^2}{x^2 + 4}$.\n\n2. **Vertical Asymptotes:** These occur where the denominator is zero and the numerat

1. **Problem Statement:** Analyze the function $$f(x) = \frac{x^2}{x^2 + 4}$$ for domain, range, first and second derivatives, intervals of increase/decrease, concavity, inflection

1. **State the problem:** Find the limit of the function $$\frac{x^2 - 1}{x - 1}$$ as $x$ approaches 1. 2. **Recall the formula and rules:** The limit of a function as $x$ approach

1. **State the problem:** Evaluate the limit $$\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$$. 2. **Recall the formula and rules:** The expression is a rational function. Direct substituti

1. **State the problem:** We need to solve the integral $$\int \frac{x^3+8}{(x^2-1)(x-2)} \, dx.$$\n\n2. **Rewrite the denominator:** Note that $$x^2-1 = (x-1)(x+1),$$ so the integ

1. **State the problem:** We need to solve the integral $$\int \frac{\sqrt{1+4x^2}}{x} \, dx.$$\n\n2. **Recall the formula and substitution:** This integral involves a square root

1. The problem is to understand what a local minimum is in the context of a function. 2. A local minimum of a function $f(x)$ is a point $x = c$ where $f(c)$ is less than or equal

1. **Problem statement:** Identify the local maxima of the function $f$ on the interval $(0,8)$ given the graph description. 2. **Recall the definition:** A local maximum at $x=c$

1. **State the problem:** We need to evaluate the improper integral $$\int_1^{\infty} \frac{1}{x^2 \sqrt{x}} \, dx$$. 2. **Rewrite the integrand:** Recall that $$\sqrt{x} = x^{1/2}

1. **State the problem:** We want to find the integral $$\int \frac{3x+1}{(x^2 + x + 1)^2} \, dx.$$\n\n2. **Recall the formula and approach:** For integrals of the form $$\int \fra

1. **State the problem:** We need to evaluate the definite integral $$\int_0^{\frac{\pi}{2}} x \sin x \, dx$$. 2. **Formula and method:** To solve this integral, we use integration

1. **State the problem:** We need to solve the integral $$\int \frac{x^3}{x^4 + 1} \, dx$$. 2. **Identify the formula and approach:** Notice that the denominator is $x^4 + 1$ and t

1. **Problem Statement:** Find the slope of the tangent line to the function $$y = (2\sqrt{x} + 1)(x^3 - 6)$$ at $$x = 0$$. 2. **Formula and Rules:** The slope of the tangent line

1. **State the problem:** We need to solve the integral $$\int x^2 \ln x \, dx$$. 2. **Formula and method:** We will use integration by parts, which states:

1. **نص المسألة:** لدينا الدالة $$f(x) = x + 1 - \frac{e^x}{e^x - 1}$$ ونريد حساب الحدود التالية: - $$\lim_{x \to 0^-} f(x)$$

1. **State the problem:** Find the volume of the solid formed by rotating the region in the first quadrant bounded by $y=\sqrt{9-x^2}$ and $y=x$ about the $y$-axis. 2. **Identify t

1. **State the problem:** Find the limit $$\lim_{x \to 2} \frac{x^2 - 3x + 2}{x^2 - 4}$$. 2. **Recall the formula and rules:** To find limits involving rational functions, first tr

1. We are asked to find the limit: $$\lim_{x \to 4} \frac{\sqrt{4 + x} - \sqrt{2x}}{x - 4}$$ 2. This is an indeterminate form of type $\frac{0}{0}$ because substituting $x=4$ gives