∫ calculus

1. **State the problem:** Evaluate the improper integral $$\int_{0}^{\infty}\frac{x\left(e^{x}-1\right)}{e^{x}\left(e^{x}-x\right)}dx.$$ 2. **Analyze the integrand:** The integrand

1. **State the problem:** We need to evaluate the volume $V$ given by the integral $$V=\int_{24}^{55} \pi \left[12 - \left(3.8 \sec^{-1}(1.02x - 19.0957) + 6\right)\right]^2 dx.$$

1. **Stating the problem:** We want to find the integral $$\int \sqrt{\sin(x)} \, dx$$. 2. **Understanding the integral:** This integral involves the square root of the sine functi

1. **State the problem:** Evaluate the integral $$\int \sum_{n=1}^{\infty} \frac{n}{2025 x^n} \, dx.$$\n\n2. **Rewrite the sum inside the integral:** The sum is $$\sum_{n=1}^{\inft

1. **State the problem:** We need to find the integral $$\int \frac{2x}{x^2+1} \, dx$$. 2. **Recall the formula and rules:** The integral of a function of the form $$\frac{f'(x)}{f

1. **Problem statement:** (i) Show that $\frac{d}{dx}(\sec x) = \sec x \tan x$ by writing $\sec x$ as $(\cos x)^{-1}$.

1. **State the problem:** We are given parametric equations $x = t^{2} + 2$ and $y = t^{2} + 9t$. We need to find the first derivative $\frac{dy}{dx}$, the second derivative $\frac

1. **State the problem:** Find the equation of the tangent line to the parametric curve defined by $$x = t^2 - 4t$$

1. **Problema planteado.** Queremos escribir correctamente la serie de Fourier y verificar sus coeficientes.

1. **State the problem:** We need to solve the differential equation $$\frac{dy}{dx} = (64xy)^{1/3}$$. 2. **Rewrite the equation:** The equation can be written as $$\frac{dy}{dx} =

1. **State the problem:** We are given the implicit equation $$-\frac{t^2}{y^2} - \cos(6 y) = -4$$

1. **State the problem:** We want to find the derivative of the function $$f(y) = 9\tan(y) - \frac{\sqrt{y}}{8} - \frac{6}{y^2\sqrt{y}}.$$ 2. **Recall derivative formulas:**

1. **State the problem:** We need to find the derivative of the function $$f(y) = \frac{16 \cdot y^{1/4}}{\ln(y)}.$$ 2. **Recall the formula:** To differentiate a quotient $$\frac{

1. **State the problem:** Find the derivative of the function $$f(y) = \frac{16 \cdot y^{\frac{1}{4}}}{\ln(y)}$$ with respect to $y$. 2. **Recall the formula:** To differentiate a

1. **State the problem:** Find the derivative of the function $$f(y) = 9 \tan(y) - \frac{\sqrt{y}}{8} - \frac{6}{y^{2}\sqrt{y}}.$$ 2. **Rewrite the function for clarity:**

1. **State the problem:** We want to evaluate the integral $$\int_0^5 \sqrt{25 - x^2} \, dx$$ using the substitution $x = 5 \sin u$ and verify the result matches the quarter circle

1. **State the problem:** We need to evaluate the integral $$\int 6 \sin 2x \cos x \, dx$$. 2. **Recall relevant formulas:** Use the product-to-sum identity for sine and cosine:

1. The problem is to understand the definite integral $\int_a^b f(x) dx$. 2. The definite integral $\int_a^b f(x) dx$ represents the signed area under the curve $y=f(x)$ from $x=a$

1. **Stating the problem:** We want to understand and evaluate the expression involving the integral and the summation: $$\int \sum_{i=1}^n i^2$$.

1. **Problem Statement:** Find the maximum, minimum, relative maximum, and relative minimum values of the function $$f(x) = \sin(x^2)$$ over the interval $$(0, 4\pi)$$. 2. **Formul

1. **State the problem:** We need to evaluate the definite integral $$\int_0^2 4x e^{2x} \, dx$$. 2. **Formula and method:** This integral involves a product of a polynomial and an