∫ calculus

1. **State the problem:** Find the derivative $y'$ of the function $$y = \ln(\cos(x^2 + 1)).$$ 2. **Recall the formula:** The derivative of $\ln(u)$ with respect to $x$ is $$\frac{

1. **State the problem:** Find the average value of the function $g(x) = 1 - 6x^2$ on the interval $[-1, 3]$. 2. **Formula for average value:** The average value of a function $f(x

1. **Planteamiento del problema:** Queremos aproximar $\sin(0.12)$ usando un polinomio de Taylor de orden 2.

1. **State the problem:** We need to find values of $a$ and $b$ such that the function $$g(x) = \begin{cases} a^2 x + 2 & \text{if } x > 3 \\ 5 & \text{if } x = 3 \\ x^2 - b x + a

1. **State the problem:** Find the limit as $x$ approaches 3 of the function $$\frac{x^2 - 9}{x - 3}$$. 2. **Recall the formula and rules:** The limit of a rational function where

1. **State the problem:** Find the limit as $x$ approaches 2 of the function $$\frac{x^2 - 4}{x - 2}$$. 2. **Recall the formula and rules:** The direct substitution of $x=2$ gives

1. **Problem 7:** Find the minimum value of the function $g(x) = x^5 - 5x^3 - 20x$ on the interval $[0,3]$. 2. To find the minimum, we first find the critical points by computing t

1. **State the problem:** Find the limit as $x$ approaches 0 of the expression $$\frac{x^2 - x}{\sqrt{3} - \sqrt{3} - x}.$$ 2. **Simplify the denominator:** Notice that $\sqrt{3} -

1. **State the problem:** Differentiate the function $$f(x) = \frac{\ln x}{x^3}$$. 2. **Rewrite the function:** Using the property $$\frac{1}{x^3} = x^{-3}$$, rewrite the function

1. **Problem:** Find the area between $y = e^x$ and $y = 1$ over the interval $[0,2]$. 2. **Formula:** The area between two curves $y = f(x)$ and $y = g(x)$ over $[a,b]$ is given b

1. **State the problem:** We need to find the derivative of the function $f(x) = 3x^2 - 7x + 1$ using the limit definition of the derivative: $$f'(x) = \lim_{h \to 0} \frac{f(x+h)

1. **Problem:** Evaluate the integral $$\int \frac{x^2 + 3x + 1}{(x+2)(x-3)^2 (x^2 + 4)^2} \, dx$$ 2. **Formula and rules:** For rational functions with polynomial denominators, us

1. **Problem Statement:** We have a piecewise function \( j(x) \) defined as:

1. **State the problem:** Find the area enclosed by the curves given by the equations: $$2y=5\sqrt{x}, \quad y=5, \quad 2y+3x=8$$

1. **Problem:** Find the derivative of $f(x) = (6x^2 + 7x)^4$ using the chain rule. 2. **Formula:** The chain rule states that if $f(x) = (g(x))^n$, then

1. **Stating the problem:** We want to create an integration system in Desmos that fits a polygon under any function, such as $f(x)=mx+b$ or $f(x)=x^2$, to approximate the area und

1. **State the problem:** Evaluate the expression $$\int_0^3 1 \, dx + \lim_{n \to \infty} \left(3 - \frac{1}{n}\right) - 3 \over \cos(0)$$. 2. **Evaluate the integral:** The integ

1. Тодорхойл: Функцийн өсөх, буурах завсарыг олох нь функцийн графикийн ямар хэсэгт функц өсч, ямар хэсэгт буурч байгааг тодорхойлох явдал юм. 2. Функцийн өсөх, буурах завсарыг оло

1. The problem is to simplify the integrand $$\frac{\csc^2 x \sin x - \sin x}{\cos x}$$ and understand why it becomes $$\frac{1 - \sin^2 x}{\sin x \cos x}$$. 2. Recall that $$\csc

1. Problem: Find the derivative of $f(x) = \frac{3}{4}$ using the constant rule. The constant rule states that the derivative of a constant is zero.

1. نبدأ بكتابة المسألة: حساب التكامل من الفترة $-1$ إلى $3$ للدالة $2-5x$ باستخدام التعريف العام للتكامل. 2. التعريف العام للتكامل هو حساب نهاية مجموع ريمان: