∫ calculus

1. The problem asks us to analyze the function $f$ based on the graph of its second derivative $f''(x)$. 2. Given that $f''(0) = 4 > 0$, the second derivative is positive at $x=0$.

1. The problem is to find the limit \( \lim_{x \to 5} (2x^5 - 3x + 4) \). 2. Since the function \(2x^5 - 3x + 4\) is a polynomial, it is continuous everywhere, so we can directly s

1. **Stating the problem:** We analyze the curve of $\hat{f}$ and determine which of the given statements about the function $f$ are correct or incorrect. 2. **Understanding the re

1. The problem is to find the limit of the function $2x^5 - 3x + 4$ as $x$ approaches 5. 2. Substitute $x = 5$ directly into the function since it is a polynomial and continuous ev

1. The problem states that the top-right graph represents the first derivative $y' = f'(x)$ of a function $y = f(x)$ defined on $\mathbb{R}$. We need to identify which of the given

1. The problem asks us to analyze the function on the interval $[0,2[$ and determine whether it has absolute minimum and/or maximum values. 2. From the graph description, the funct

1. **State the problem:** Find the indefinite integral $$\int \frac{x^2 - 4}{x + 2} \, dx.$$\n\n2. **Simplify the integrand:** Notice that $$x^2 - 4$$ can be factored as $$(x - 2)(

1. The problem states that $$\int \frac{2}{y} \, dy = \int \frac{1}{x} \, dx$$ and asks to find the expression for $$\ln y^2$$ in terms of $$x$$ plus a constant $$c$$. 2. Compute t

1. **State the problem:** We need to evaluate the integral $$\int \frac{dx}{\sqrt{x}(\sqrt{x} + 2)^4} + c.$$\n\n2. **Substitution:** Let $$t = \sqrt{x} + 2.$$ Then $$\sqrt{x} = t -

1. **Stating the problem:** We want to find the integral $$\int 4 \sin^4 x \, dx$$ and match it with one of the given options. 2. **Rewrite the integral:**

1. **State the problem:** We need to evaluate the integral $$\int \frac{3}{\sin^2 3x} \, dx$$ and match it with one of the given options. 2. **Rewrite the integrand:** Recall that

1. The problem is to evaluate the integral $$\int \frac{\ln x}{x} \, dx$$ and identify the correct form of the antiderivative from the given options. 2. Let us use substitution to

1. The problem is to evaluate the integral $$\int \frac{6}{x} (\ln x)^5 \, dx$$ and match it with one of the given options. 2. Notice that the integral involves a function of $\ln

1. The problem asks to find the indefinite integral $$\int e^2 \, dx$$ plus a constant of integration $c$. 2. Note that $e^2$ is a constant because it does not depend on $x$.

1. The problem is to find the integral $$\int a^{3^{\log_a x}} \, dx$$ plus the constant of integration $c$. 2. First, simplify the exponent: note that $$3^{\log_a x}$$ is a bit un

1. The problem is to find the limit of a function as the variable approaches a certain value. 2. Since the user only wrote "lim" without specifying the function or the point of app

1. **State the problem:** Evaluate the integral $$\int e^{\ln x} \, dx$$ and identify the correct answer from the options. 2. **Simplify the integrand:** Recall that $$e^{\ln x} =

1. The problem is to analyze the piecewise function: $$f(x) = \begin{cases} -e^{\frac{1}{x}}, & x < 0 \\ \ln\left(\frac{1}{1+x^2}\right), & x > 0 \end{cases}$$

1. **المطلوب:** حساب مشتقات الدوال التالية: 2. **المطلوب:** حساب حدود الدوال باستخدام قاعدة لوبتال.

1. **تمرين 02: حساب التكاملات** 1. \(\int x\sqrt{x} \, dx = \int x x^{\frac{1}{2}} \, dx = \int x^{\frac{3}{2}} \, dx = \frac{2}{5} x^{\frac{5}{2}} + C\).

1. The problem discusses the concept of the derivative of a function $f$ at a point $a$ from the right and from the left. 2. The right-hand derivative at $a$ is defined as $$f'(a^+