∫ calculus

1. Problem: Find the derivative of the function $f(x) = \frac{2}{3} x^{-3}$.\n\n2. Recall the power rule for derivatives: If $f(x) = ax^n$, then $f'(x) = a n x^{n-1}$.\n\n3. Apply

1. Problem: Find the derivative of the function $f(x) = -5 \sqrt[3]{x^2}$. 2. Rewrite the function using exponents: $f(x) = -5 x^{\frac{2}{3}}$.

1. We are given the function $$f(t) = 5t^2 - \frac{1}{t^3} + 3t - \sqrt{t} + 1$$ and asked to find the second derivative $$f''(t)$$ at $$t=1$$. 2. First, recall the rules for deriv

1. **State the problem:** Find the definite integral $$\int_0^1 \frac{t^2 + 1}{t^4 + 1} \, dt.$$\n\n2. **Recall the formula and approach:** We want to integrate a rational function

1. The problem asks for the value of $f(5)$ and the limits of $f(x)$ as $x$ approaches 5 from the left, right, and both sides. 2. From the graph description, near $x=5$, the bottom

1. **State the problem:** We want to evaluate the integral $$\int_0^{1} \frac{x^x (x^{2x} + 1) (\ln x + 1)}{x^{4x} + 1} \, dx.$$\n\n2. **Analyze the integrand:** The integrand is $

1. **State the problem:** Find the second derivative $f''(t)$ of the function $$f(t) = 5t^2 - \frac{1}{t^3} + 3t - \sqrt{t} + 1$$ and then evaluate it at $t=1$. 2. **Rewrite the fu

1. **Problem (a):** Find the gradient of the curve $y = 3x^4 - 2x^2 + 5x - 2$ at points $(0, -2)$ and $(1, 4)$. The gradient of a curve at any point is given by the derivative $\fr

1. We are asked to find the limit: $$\lim_{x \to +\infty} \left(1 - \frac{3}{x}\right)^x$$ 2. This is a classic limit of the form $$\lim_{x \to \infty} \left(1 + \frac{a}{x}\right)

1. **Problem statement:** Find the limit $$\lim_{x \to +\infty} \left(1 - \frac{3}{x}\right)^x$$. 2. **Recall the formula:** The limit $$\lim_{x \to \infty} \left(1 + \frac{a}{x}\r

1. The problem is to find the derivative of the function $f(x) = x^2$. 2. The formula for the derivative of a power function $f(x) = x^n$ is given by the power rule: $$\frac{d}{dx}

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:**