∫ calculus

1. The problem states that for the function $f$, at $x=5$, we have $f(5)=7$, $f'(5)=0$, and $f''(5)=-4$. We need to determine the nature of the point $(5,7)$. 2. Since $f'(5)=0$, t

1. The problem asks for the number of critical points of the function $f(x) = x^3 - 3x$ defined on the interval $]-1,4[$. 2. Critical points occur where the derivative $f'(x)$ is z

1. **Problem statement:** Given that $f$ is an increasing function on its domain, determine which of the following functions must also be increasing on their domains: (a) $y = f(x)

1. **State the problem:** Determine the intervals where the function $$f(x) = x^3 + 4x + 2$$ is increasing. 2. **Find the derivative:** The function is increasing where its derivat

1. **State the problem:** We want to find the intervals where the function $f(x) = \frac{x}{x^2+1}$ is increasing. 2. **Find the derivative:** To determine where $f(x)$ is increasi

1. **הגדרת הבעיה:** נתונה הפונקציה $$f(x) = \frac{e^{2x} - 9e^x}{e^{2x} - 10e^x + 9}$$

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

1. We are given the integral $$\int \frac{4x^3 - ax}{x^4 - 2x^2 + 3} \, dx = \ln|x^4 - 2x^2 + 3| + C$$ and need to find the value of $a$. 2. Notice that the derivative of the denom

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. We are asked to find the integral $$\int \frac{\tan x}{\cos x} \, dx$$ and identify which of the given options matches the result. 2. Recall that $$\tan x = \frac{\sin x}{\cos x

1. The problem is to evaluate the integral $$\int \cos(\tan x + 1) \sec^2 x \, dx.$$\n\n2. Notice that the integrand contains $\cos(\tan x + 1)$ and $\sec^2 x$. Recall that the der

1. We are asked to evaluate the integral $$\int \frac{\cos 2x}{\cos x + \sin x} \, dx$$ and then match the result with one of the given options. 2. Start by simplifying the denomin

1. **State the problem:** Evaluate the integral $$\int (\sin 3x \cos x - \cos 3x \sin x) \, dx$$. 2. **Recognize the trigonometric identity:** The expression inside the integral ma

1. The problem is to find the integral $$\int \frac{\ln x}{x} \, dx$$ and match it with one of the given options. 2. Let us use substitution to solve the integral. Set $$u = \ln x$

1. We are asked to evaluate the integral $$\int \frac{6}{x} (\ln x)^5 \, dx$$ and identify the correct expression from the given options. 2. Notice that the integral involves a fun

1. We are asked to evaluate the integral $$\int \sin x e^{\cos x} \, dx.$$\n\n2. Notice that the integrand contains $\sin x$ and $e^{\cos x}$. We can try substitution. Let $$u = \c

1. **State the problem:** We need to find the indefinite integral $$\int x^5 e^{x^6 + 1} \, dx$$. 2. **Identify substitution:** Let $$u = x^6 + 1$$. Then, differentiate:

1. **State the problem:** Simplify the integral $$\int \frac{x}{x+1} \, dx$$. 2. **Rewrite the integrand:** Notice that $$\frac{x}{x+1} = \frac{x+1-1}{x+1} = \frac{x+1}{x+1} - \fra

1. **State the problem:** We want to find the integral $$\int \ln(x+1) \, dx$$ using integration by parts. 2. **Recall integration by parts formula:** $$\int u \, dv = uv - \int v

1. **State the problem:** We need to find the limit $$\lim_{x \to 0} \frac{x^5 + 2x^2}{e^x - x - 1}$$. 2. **Analyze the expression:** As $x \to 0$, both numerator and denominator a

1. **State the problem:** We want to find the limit $$\lim_{x \to 2} \frac{2 - x}{\sqrt{x+5} - \sqrt{5}}.$$\n\n2. **Rationalize the denominator:** Multiply numerator and denominato