∫ calculus

1. The problem is to find the derivative of the function $y = x^2$ with respect to $x$. 2. The formula for the derivative of a power function $y = x^n$ is given by:

1. **State the problem:** We are given the function $$f(x) = 4(3x - 4)^{-1} + 3x$$ for $$x \geq \frac{3}{2}$$ and need to find the stationary value at $$x = a$$, i.e., find $$a$$ w

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{\sqrt{x+4} - 2}{x}$$. 2. **Recall the formula and rules:** When direct substitution leads to an indeterminate form l

1. **State the problem:** We are given the curve equation $$y = k(3x - k)^{-1} + 3x$$ where $k$ is a constant. We need to find the values of $x$ at which the curve has stationary p

1. Το πρόβλημα ζητά να υπολογίσουμε το ολοκλήρωμα του $\cos^4 x$ ως προς $x$. 2. Χρησιμοποιούμε τον τύπο μείωσης δυνάμεων για το συνημίτονο: $$\cos^2 x = \frac{1 + \cos 2x}{2}$$

1. **Problem statement:** Find the indefinite integral of the function

1. **Problem statement:** Evaluate the integral $$I = \int \sqrt{1 + \cos x} \, dx$$. 2. **Formula and trigonometric identity:** Use the half-angle identity for cosine: $$1 + \cos

1. **Problem Statement:** Prove that if the limit of a function $f(x)$ as $x$ approaches $x_0$ exists, then this limit is unique. 2. **Definition of Limit:** The limit of $f(x)$ as

1. **Problem:** Use the Weierstrass substitution method to evaluate the integral $$\int \frac{\sin x}{1 + \cos x} \, dx.$$\n\n2. **Recall the Weierstrass substitution:** Let $$t =

1. **Problem Statement:** Evaluate the integral $$\int \sqrt{1 + \sin x} \, dx$$. 2. **Formula and Identities Used:**

1. **State the problem:** We want to find the limit as $x$ approaches positive infinity of the expression $$\left( \frac{\sqrt[x]{2} + \sqrt[x]{4} + \sqrt[x]{8}}{3} \right)^x.$$\n\

1. **State the problem:** Find the value of $a$ such that the limit $$\lim_{x \to 0} \frac{a \sin x - \sin 2x}{\tan^3 x}$$ is finite. 2. **Recall important formulas and approximati

1. **State the problem:** Find the limit as $x \to \infty$ of $$\left(\frac{\sqrt[3]{2} + \sqrt[3]{4} + \sqrt[3]{8}}{3}\right)^2$$. 2. **Understand the expression:** The expression

1. Differentiate $y = x \sin x$ with respect to $x$. Use the product rule: $\frac{d}{dx}[uv] = u'v + uv'$ where $u = x$ and $v = \sin x$.

1. **State the problem:** Find the derivative $y'$ if $y = -e^{\csc(x^2)}$. 2. **Recall the chain rule:** If $y = f(g(x))$, then $y' = f'(g(x)) \cdot g'(x)$.

1. **State the problem:** Evaluate the integral $$\int e^{\tan\theta} (\sec\theta - \sin\theta) \, d\theta.$$\n\n2. **Recall relevant formulas and rules:** We will use substitution

1. The problem involves the expression $$\oint 2xy \frac{dy}{dx} + 2y^2 - 3x - d$$ which appears to be a line integral or a differential expression involving $x$, $y$, and their de

1. **Problem Statement:** Given the graph of the derivative function $f'(x)$ for $-1 \leq x \leq 3$, determine the number of inflection points of the original function $f(x)$. 2. *

1. The problem asks to find the interval where the derivative $f'$ of the function $f$ is negative. 2. The derivative $f'$ is negative where the function $f$ is decreasing.

1. The problem asks us to identify which graph could represent the second derivative $f''(x)$ of a function $f(x)$ given its shape. 2. The original function $f(x)$ starts below the

1. The problem involves identifying the derivative function $f'(x)$ of a cubic function $f(x)$ shown in the top-right graph. 2. The top-right graph shows $f(x)$, a cubic curve incr