∫ calculus

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

1. Let's state the problem: We want to understand which Riemann sum, left or right, generally overestimates the integral of a function. 2. The formula for the left Riemann sum over

1. **Problem Statement:** Calculate the LEFT(2), RIGHT(2), and MID(2) Riemann sums for the integral $$\int_1^2 \frac{1}{x} \, dx$$ using 2 subdivisions, and compare these approxima

1. **Problem statement:** We have the function $f(x) = \frac{1}{x} + \frac{\ln x}{x}$ defined for $x > 0$. We want to prove that the vertical line $x=0$ is an asymptote of the curv

1. **Problem statement:** (i) Find the derivative $\frac{dy}{dx}$ if $y = \ln(2x^2 + 5)$.

1. **Problem 10.1:** Use the Mean Value Theorem (MVT) to show that $$\cos b - \cos a \leq b - a$$ for all real numbers $$a < b$$. 2. **Mean Value Theorem statement:** If a function

1. **State the problem:** We are given the differential equation $y' = \frac{x^3}{x^2 + 1}$ and need to find the function $y(x)$. 2. **Recall the formula:** Since $y' = \frac{dy}{d