∫ calculus

1. **State the problem:** Determine whether the integral $$\int_0^1 \frac{x}{x-1} \, dx$$ converges or diverges. 2. **Analyze the integrand:** The integrand is $$f(x) = \frac{x}{x-

1. **State the problem:** Derive the equation $x^2 + 3x + 4 = 2y^2$ implicitly to find $\frac{dy}{dx}$ and then find the equation of the normal line at the point $(1, 2)$. Also, co

1. **State the problem**: We want to show using the first principle of differentiation that $$\frac{d}{dx}(\cos x) = -\sin x.$$\n\n2. **Recall the definition of derivative from fir

1. Find $\int x^2 \, dx$. 2. Evaluate $\int e^x \, dx$.

1. **State the problem:** We want to find the derivative of the function $f(x) = \frac{1}{x-2}$ using the first principle (definition of derivative). 2. **Recall the definition of

1. Integration is a fundamental concept in calculus that involves finding the integral of a function. 2. The integral represents the area under the curve of a function over an inte

1. The problem is to show the derivative of a function $f(x)$ using first principles, which means using the definition of the derivative as a limit. 2. By definition, the derivativ

1. **Stating the problem:** Evaluate the limit $$\lim_{x\to \infty} \left(1 + \frac{5}{3x}\right)^{\frac{x}{2}}.$$\n\n2. **Rewrite the expression:** Notice the limit has the form s

1. We are asked to evaluate the limit $\lim_{x \to \infty} \left(1 + \frac{5}{3x}\right)$.\n\n2. As $x$ approaches infinity, the term $\frac{5}{3x}$ approaches $0$ because the deno

1. Problem a: Find $$\lim_{n \to \infty} \sqrt[n]{5n + 3}.$$ We rewrite as $$ (5n+3)^{1/n} = e^{\frac{1}{n} \ln(5n+3)}.$$

1. Stating the problem: Evaluate the limit $\lim_{n \to \infty} n \sqrt{5n + 3}$. 2. Rewrite the expression inside the limit:

1. We are asked to find the derivative $f'(x)$ of the function $f(x) = 4x + 7$ using the definition of the derivative. 2. The definition of the derivative is:

1. We need to find the derivative $f'(x)$ of the function $f(x) = 4x + 7$ using the definition of the derivative. 2. The definition of the derivative at a point $x$ is given by:

1. **State the problem:** We want to find the limit $$\lim_{x \to \infty} \frac{5^{x+1} + 7^{x+1}}{5^x - 7^x}$$ without using differentiation or L'Hopital's Rule.

1. **State the problem:** We are given three functions: $$f_1(x) = 2x^{6} \sin(x), \quad f_2(x) = x^{5} \cdot 5^{x}, \quad f_3(x) = x^{5} \ln(x) \cos(x)$$

1. **State the problem:** We have the parabola given by $$y = (9 - x)(x - 3)$$ and the horizontal line $$y = k - 3$$ where $$k > 3$$. We need to find the area of the shaded region

1. **Problem Statement:** (i) Find the integral $\int \sqrt{4 + x} \, dx$ without a calculator.

1. State the problem: Find the limit $$\lim_{x \to 7} (2x + 2)$$. 2. Since the function $2x + 2$ is a polynomial (which is continuous everywhere), the limit as $x$ approaches 7 is

1. The problem gives a differential equation: $$\frac{d w}{d x} = -kW^{3} T^{-\frac{1}{2}}.$$\n\n2. We want to solve this differential equation for $w$ as a function of $x$. Howeve

1. Stated Problem: Differentiate the function $$H(x) = 3 \sec x (1 - \tan x)$$. 2. Use the product rule for differentiation: If $$H(x) = f(x)g(x)$$, then $$H'(x) = f'(x)g(x) + f(x)

1. Stating the problem: We are asked to find two limits based on the description of the graph of a function $f(x)$. 2. For part (a), find $\lim_{x \to 0} f(x)$.