∫ calculus

1. **State the problem:** Find $\frac{dx}{dy}$ given the implicit equation $$x = \sqrt{xy} + 3$$.

1. **Problem Statement:** Find the derivative $\frac{dy}{dx}$ using implicit differentiation for an equation involving both $x$ and $y$. 2. **Formula and Rules:** When differentiat

1. Differentiate $$y^3 = 2xy^2 + x^2y$$ with respect to $$x$$. Use implicit differentiation and product rule.

1. **State the problem:** Differentiate the equation $$\frac{y}{x} = \sqrt{xy} + 3$$ with respect to $$y$$. 2. **Rewrite the equation:**

1. **State the problem:** We are given the function $f(x) = 3e^x + 10 \ln(x)$ and need to find the derivative evaluated at $x=3$, i.e., $f'(3)$. 2. **Recall the derivative rules:**

1. **State the problem:** Find the derivative of the function $$y = x^2$$ at the point $$x = 3$$ using first principles.

1. **State the problem:** Calculate the value of

1. **State the problem:** Differentiate the equation $$\frac{y}{x} = \sqrt{xy} + 3$$ with respect to $$y$$. 2. **Rewrite the equation:**

1. **Problem:** Differentiate implicitly with respect to $x$ the equation $$y^3 = 2xy^2 + x^2y.$$ 2. **Formula and rules:** Use implicit differentiation. Recall the product rule: $

1. **Problem:** Evaluate $\int \sin 2\theta (\sin^2 \theta + 5)^4 d\theta$. **Step 1:** Recognize the chain rule pattern. Let $u = \sin^2 \theta + 5$.

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 $$-\frac{t^2}{y^2} - \cos(6 y) = -4$$