∫ calculus

1. **State the problem:** Find the derivative of the function $$y = -8x^{-8} + 12\sqrt{x}$$ using the power rule. 2. **Recall the power rule:** For any function $$y = x^n$$, the de

1. **State the problem:** We need to determine the integral of the function $$\frac{x^3+5}{x^2-25}$$ with respect to $x$. 2. **Recall the formula and rules:** To integrate a ration

1. **State the problem:** We need to determine the derivative of the function $$f(x) = \frac{x^3 + 5}{x^2} - 25$$ with respect to $$x$$. 2. **Rewrite the function:** To differentia

1. **Problem statement:** Find the indefinite integral $$\int \frac{1}{t^4} \, dt$$. 2. **Rewrite the integrand:** Recall that $$\frac{1}{t^4} = t^{-4}$$.

1. **Problem statement:** (a) Given that $|f(x) - 3| \leq 4(x - 2)^2$, find $\lim_{x \to 2} f(x)$.

1. **Problem statement:** Find the limit of the function $$\frac{\ln x}{10\sqrt{x}}$$ as $$x \to \infty$$ using the derivative ratio $$\frac{g'(x)}{h'(x)}$$ where $$g(x) = \ln x$$

1. **State the problem:** We have a function $f$ with an inverse $f^{-1}$ defined on an interval $I$. Given $f(3) = -4$ and the derivative of the inverse at $-4$ is $(f^{-1})'(-4)

1. **State the problem:** We are given the inequality $|f(x) - 3| \leq 4(x - 2)^2$ and asked to find $\lim_{x \to 2} f(x)$. 2. **Recall the squeeze theorem:** If $g(x) \leq f(x) \l

1. **Problem (a):** Given that $|f(x) - 3| \leq 4(x - 2)^2$, find $\lim_{x \to 2} f(x)$. 2. **Step 1:** Understand the inequality. The expression $|f(x) - 3| \leq 4(x - 2)^2$ means

1. **Problem statement:** (a) Given that $|f(x) - 3| \leq 4(x - 2)^2$, find $\lim_{x \to 2} f(x)$.

1. **Problem 1:** Evaluate the function $2\cos(2\pi t)$. 2. **Problem 2:** Evaluate the infinite series

1. Το πρόβλημα ζητά να υπολογίσουμε το ολοκλήρωμα: $$\int x^{2} \sqrt{x^{3} + 1} \, dx$$

1. Το πρόβλημα ζητά να βρούμε το όριο $$\lim_{x \to +\infty} xe^{-x}$$

1. Vamos resolver uma integral onde o integrando é uma função mais complicada que 1, contendo incógnitas. 2. Suponha que queremos calcular a integral $$\int (3x^2 + 2x + 1) \, dx$$

1. **State the problem:** We want to find the derivative of the function $y = x + 3$ using the first principle of derivatives. 2. **Recall the formula for the derivative using firs

1. The problem is to evaluate the definite integral $\int_2^2 3x \, dx$. 2. The formula for the definite integral of a function $f(x)$ from $a$ to $b$ is:

1. The problem is to evaluate the definite integral $\int_2^2 3x \, dx$. 2. The formula for the definite integral of a function $f(x)$ from $a$ to $b$ is:

1. The problem involves understanding and comparing two functions: $$f(t) = \cos(at)$$ and $$f(t) = \sinh(at)$$. 2. The cosine function $$\cos(at)$$ is a periodic function with per

1. **State the problem:** We need to evaluate the double integral

1. **State the problem:** We need to evaluate the double integral $$\int_0^\infty \int_0^{\frac{\pi}{2}} \frac{x \sin \theta \ln(1 + x^2 \cos^2 \theta)}{(1 + x^2 \sin^2 \theta)^{3/

1. **State the problem:** We are given the position function of an object as $x(t) = t e^{-at}$ where $a$ is a positive constant and $t > 0$. 2. **Understand the function:** This f