∫ calculus

1. **Problem Statement:** Find the stationary points of a function, which are points where the derivative equals zero. 2. **Formula and Rules:** Stationary points occur where the f

1. **Problem Statement:** Find the local maximum of the function $$f(x) = \frac{1}{1 + x^2}$$ over the domain $$D = [-1, 3]$$. 2. **First Order Condition (FOC):** To find critical

1. The problem is to evaluate the definite integral $$\int_{0}^{\pi/3} \frac{\sin(x)}{\cos^2(x)} \, dx$$. 2. We use substitution to simplify the integral. Let $$u = \cos(x)$$. Then

1. **Problem 1: Find the dimensions of a rectangle with area 1000 m² that minimize the perimeter.** 2. Given:

1. **Problem Statement:** We are given the infinite series $$\sum_{n=1}^{\infty} \frac{(3x - 2)^n}{n}$$ and asked to find its series representation as a function. 2. **Recall the f

1. **State the problem:** We need to evaluate the definite integral $$\int_0^1 x e^{-9x^2} \, dx$$ and verify the result using a graphing utility. 2. **Recall the formula and subst

1. **Problem statement:** Find the indefinite integral $$\int e^x \sqrt{9 - e^x} \, dx$$ including the constant of integration. 2. **Formula and substitution:** To solve integrals

1. **State the problem:** Find the indefinite integral $$\int \frac{e^{1/x^2}}{x^3} \, dx$$ including the constant of integration. 2. **Identify a substitution:** Notice the expone

1. **State the problem:** Find the indefinite integral $$\int \frac{e^{\sqrt{5x}}}{\sqrt{5x}} \, dx$$ including the constant of integration. 2. **Identify substitution:** Let $$u =

1. **State the problem:** Find the indefinite integral $$\int e^{8x - 2} \, dx$$. 2. **Recall the formula:** The integral of an exponential function $$e^{ax+b}$$ with respect to $$

1. **State the problem:** We need to find the indefinite integral of the function $$1 + t^2 + t^4 + t^6$$ with respect to $$t$$. 2. **Recall the integral formula:** The integral of

1. **Problem Statement:** Given the graph of a function $f$ from $x=0$ to $x=6$, determine the intervals where $f$ is increasing, decreasing, concave upward, concave downward, and

1. **State the problem:** We need to find the indefinite integral of the function $$7r^5 + 4r^2 + 1$$ with respect to $$r$$. 2. **Recall the integral formula:** The integral of $$r

1. **Problem Statement:** We are given a function $y = g(x)$ with information about its absolute and local extrema. 2. **Definitions:**

1. The problem is to evaluate the integral $$\int \frac{7}{2x^{9/4}} \, dx$$. 2. Rewrite the integrand using properties of exponents: $$\frac{7}{2x^{9/4}} = \frac{7}{2} x^{-9/4}$$.

1. The problem is to find the indefinite integral of the function $3x^{37}$ with respect to $x$. 2. The formula for integrating a power function $x^n$ is:

1. **State the problem:** We want to find the limit as $n$ approaches infinity of the expression $$\sqrt{2n+3} - \sqrt{n-1}.$$\n\n2. **Recall the formula and technique:** When deal

1. **State the problem:** We want to find the limit $$\lim_{n \to \infty} \frac{\sqrt{n^2+1} + n}{\sqrt[4]{n^3 + n} - \sqrt{n}}$$

1. **State the problem:** We want to find the limit as $n \to \infty$ of the expression $$\frac{\sqrt{n^2+1}+n}{\sqrt[4]{n^3+n}-\sqrt{n}}.$$\n\n2. **Recall important rules:** When

1. Problem 5: Find $$\lim_{y \to 2^-} \frac{(y-1)(y-2)}{y+1}$$. Formula: For limits involving rational functions, if direct substitution leads to a defined value, that is the limit

1. **Problem Statement:** Determine for which values of the exponent $p$ the improper integral $$\int_1^\infty \frac{1}{x^p} \, dx$$ diverges. 2. **Formula and Setup:** For $p \neq