∫ calculus

1. We are asked to evaluate the definite integral $$\int_0^1 x^2 e^{2x} \, dx$$. 2. To solve this integral, we use integration by parts. Let:

1. **State the problem:** We need to solve the differential equation $$\frac{dy}{dx} = x^2 - 2$$ with the initial condition $$y(3) = 7$$. 2. **Integrate the differential equation:*

1. The problem is to simplify the differential equation $$\frac{dy}{dx} = \frac{1}{3y}$$ and find the implicit solution. 2. Start by separating variables: multiply both sides by $$

1. **State the problem:** Evaluate the triple integral $$\int_0^1 \int_0^1 \int_0^{x+y} \ln z \, dz \, dy \, dx.$$\n\n2. **Integrate with respect to $z$: ** We first compute $$\int

1. Problem statement: A swimming pool is 20 ft wide and 40 ft long and its cross-section has shallow depth 3 ft and deepest depth 9 ft, with the cross-section vertices at $ (0,9),

1. **State the problem:** We have a swimming pool with varying depth and a trapezoidal cross-section. The pool is being filled at a rate of 0.8 ft³/min. We want to find how fast th

1. **State the problem:** We have a cost function $C(x) = x^3 - 3x^2 + 4x$ where $x$ is in hundreds of micro-components and $C(x)$ is in hundreds of dollars.

1. **State the problem:** We need to find the intervals where the function $$k(x) = \frac{1}{x^{2}+3}$$ is convex downward. 2. **Recall the definition:** A function is convex downw

1. **State the problem:** We need to find the interval where the function $$f(x) = x^3 - 6x^2 + 9x + 1$$ is convex upward (concave up). 2. **Recall the definition:** A function is

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