∫ calculus

1. **State the problem:** Find the limit $$\lim_{x \to \infty} \left(\frac{3x - 2}{3x + 9}\right)^{3 - 11x}$$. 2. **Rewrite the expression inside the limit:** Simplify the fraction

1. The problem is to find the derivative of the function $f(x) = 4x^3 + x^2 - x + 5$ using the limit definition of the derivative: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

1. Problem: Analyze the function $$y = \frac{x^3}{3} - \frac{x^2}{2} - 2x + \frac{1}{3}$$ to find local maxima, minima, inflection points, and intervals of concavity. 2. Find the f

1. **Problem:** Find the value(s) of $c$ that satisfy the Mean Value Theorem (MVT) for $f(x) = x^2 + 2x - 1$ on $[0,1]$. **Step 1:** State the MVT formula:

1. Problem: Determine if the function $y = h(x)$ has any absolute extreme values on $[a,b]$ from the given graph. Step 1: Theorem 1 states that a continuous function on a closed in

1. **Problem Statement:** Determine which of the following definite integrals can be evaluated over the given limits: (a) $$\int_{-1}^1 \frac{x+1}{x-1} \, dx$$

1. **Problem Statement:** Determine which of the following definite integrals can be evaluated: (a) $$\int_{1}^{-1} \frac{x+1}{x-1} \, dx$$

1. **Problem Statement:** Find the Laplace Transform of the function $$f(t) = t^3 \cos t$$. 2. **Recall the Laplace Transform formula:** The Laplace Transform of $$t^n \cos(at)$$ i

1. **State the problem:** Find the Laplace Transform of the function $f(t) = t^3 \cos t$. 2. **Recall the formula:** The Laplace Transform of a function $f(t)$ is defined as

1. **State the problem:** We need to evaluate the integral $$\int \frac{2x^3}{18x + 2x^3} \, dx.$$\n\n2. **Simplify the integrand:** Factor the denominator:\n$$18x + 2x^3 = 2x(9 +

1. **Πρόβλημα 19.13**: Δίνεται η συνάρτηση $$f(x) = 4e^{x - 1} + 4xe^{x - 1} - 6x - 1$$ **α)** Εξετάζουμε αν η $$f$$ ικανοποιεί τις προϋποθέσεις του θεωρήματος Bolzano στο διάστημα

1. **Δίνεται το πρόβλημα:** Να εξετάσουμε αν η συνάρτηση $$f(x) = 4e^{x-1} + 4xe^{x-1} - 6x - 1$$ ικανοποιεί τις προϋποθέσεις του θεωρήματος Bolzano στο διάστημα $$[0,1]$$. 2. **Θε

1. We are asked to find the limit: $$\lim_{x \to 2} \frac{x^2 + 4x}{x^2 - 4x}$$ 2. The formula for limits involving rational functions is to first try direct substitution. If it re

1. **Problem:** Solve the integral $$\int x \sin^2(x) \, dx$$. 2. **Formula and rules:** Use the identity $$\sin^2(x) = \frac{1 - \cos(2x)}{2}$$ to simplify the integral.

1. **State the problem:** We need to solve the integral $$\int x (\sin^2(x) - \cos^2(x)) \, dx$$. 2. **Use trigonometric identities:** Recall the identity $$\sin^2(x) - \cos^2(x) =

1. **State the problem:** We need to solve the integral $$\int 4x \cos^2(x) \, dx$$. 2. **Use the double-angle identity:** Recall that $$\cos^2(x) = \frac{1 + \cos(2x)}{2}$$.

1. The problem is to solve the integral $$\int x \cos(x) \, dx$$. 2. We use integration by parts formula: $$\int u \, dv = uv - \int v \, du$$.

1. The problem is to evaluate the integral $$\int \frac{9}{2} \sqrt{x} \ln x \, dx$$ and match it with one of the given options. 2. Rewrite the integral in a simpler form: $$\int \

1. **Problem Statement:** Evaluate the integral $$\int \sqrt{1 + \sqrt{x}} \, dx.$$\n\n2. **Substitution:** Let $$u = 1 + \sqrt{x}.$$ Then $$\sqrt{x} = u - 1$$ and $$x = (u - 1)^2.

1. **Problem:** Solve the integral $$\int \frac{3x}{x+2} \, dx$$. 2. **Step 1: Simplify the integrand.**

1. Problem 16: Evaluate the integral $$\int x^2 e^{x^3} \, dx$$. 2. To solve this, use substitution. Let $$u = x^3$$, then $$\frac{du}{dx} = 3x^2$$ or $$du = 3x^2 dx$$.