∫ calculus

1. **State the problem:** We have the curve $y = x \ln x$ and a line $2x - 2y + 3 = 0$. We want to find the equation of the perpendicular to the curve that is parallel to this line

1. **State the problem:** We have the curve $y = x \ln x$ and a line $2x - 2y + 3 = 0$. We want to find the equation of the perpendicular to the curve that is parallel to this line

1. **State the problem:** Find the area enclosed by the curves given by the equations $$y = 5x + 4$$ and $$y = x^2 + 2x - 6$$. 2. **Find the points of intersection:** Set the two e

1. The user mentioned the topics: integration, differentiation, exponential, and logarithms. 2. These are fundamental topics in calculus and mathematical analysis.

1. **State the problem:** We have the curve defined by the equation $$y=\sqrt{1-x^2}$$ and points A(0.6, 0.8), B(0.7, y_B), C(0.8, y_C), and D(0.9, y_D) on the curve. We need to ve

Problem: Determine which of the following compositions are convex or concave for $x \ge 0$: $f(x)=x^3$, $g(x)=x^2$, $h(x)=x^{1/3}$. 1. Compute $f(g(x))$.

1. The problem is to evaluate the integral $$\int_0^1 (3x^3 - 2x^2 + x - 4) x y^2 \sqrt{x^2 - 3x + 24} \, dx$$ with respect to $x$ from 0 to 1, treating $y$ as a constant. 2. First

1. **State the problem:** Differentiate the function $$f(x) = x^4 - x - 3 + \frac{1}{\sqrt{x}}$$. 2. **Rewrite the function:** Express the term $$\frac{1}{\sqrt{x}}$$ as a power of

**Problem:** Differentiate the function $$f(x) = (x^2 - x)(2x^3 - 1)^3$$. 1. **State the problem:** We need to find the derivative $$f'(x)$$ of the product of two functions: $$u(x)

1. **Problem statement:** Use linear approximation to estimate (a) $\sqrt[3]{1001}$ and (b) $8.06^{\frac{2}{3}}$. 2. **Recall linear approximation formula:** For a function $f(x)$

1. **Compute** $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$. Factor numerator: $x^2 - 4 = (x-2)(x+2)$.

1. **Problem statement:** Evaluate the following limits: (i) $$\lim_{x \to 0} \frac{\sqrt{1+x} - \sqrt{1-x}}{x}$$

1. **State the problem:** We are given the curve $$y = \frac{x - a}{(x + b)(x - 2)}$$ and a point on the curve $(1, -3)$. The equation of the normal to the curve at this point is $

1. The problem is to evaluate the integral $$\int \frac{\sin z}{(z+\Pi)x^3} \, dz$$ where the variable of integration is $z$ and $x$ is treated as a constant. 2. Since $x$ is const

1. **State the problem:** Evaluate the definite integral $$\int_{-1}^1 \frac{dx}{\sqrt[3]{9 + 4\sqrt{5} x (1 - x^2)^{2/3}}}$$

1. **Problem statement:** Prove the integral identity: $$\int_{-1}^1 \frac{dx}{\sqrt[3]{9 + 4\sqrt{5} x (1 - x^2)^{2/3}}} = \frac{3^{3/2}}{2^{4/3} 3^{5/6} \pi} \Gamma^3\left(\frac{

1. The problem is to evaluate the integral $$\int \cos(4x) \sin(2x) \, dx$$. 2. Use the product-to-sum identity for sine and cosine: $$\cos A \sin B = \frac{1}{2} [\sin(A+B) - \sin

1. The problem is to find the turning points of the function $y = x^2 - 6x + 6$. 2. To find turning points, we first find the derivative $y' = \frac{dy}{dx}$.

1. **Problem 4.2:** Determine the equation of the tangent line to the curve $f(x) = 5x^2 + 4x - 1$ at $x = 3$. 2. **Find the derivative $f'(x)$:**

1. **State the problem:** We need to find the limit $$\lim_{x \to 0} \frac{4 \tan(\frac{x}{2}) - 2 \sin x}{x^3}.$$\n\n2. **Recall series expansions:**

1. Problem: Compute the integral $$\int e^{ax+b} \, dx$$. Step 1: Recognize that the integral of an exponential function with a linear exponent is $$\int e^{ax+b} \, dx = \frac{1}{