∫ calculus

1. **State the problem:** Evaluate the integral $$\int -2x \ln(-x) \, dx$$. 2. **Recall the formula for integration by parts:**

1. **Stating the problem:** Evaluate the integral $$\int (2x + 1)^3 e^{-x/2} \, dx$$. 2. **Formula and approach:** This integral involves a product of a polynomial and an exponenti

1. **Problem:** Calculate the limit $$\lim_{x \to 5} \frac{x^2 - 2x - 15}{x^2 - 25}$$ 2. **Recall the formula:** Limits involving rational functions can often be simplified by fact

1. **Problem:** Evaluate the integral $$\int_0^{\frac{\pi}{3}} \sec^2 \theta \, d\theta$$. 2. **Formula and rules:** Recall that the derivative of $\tan \theta$ is $\sec^2 \theta$.

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{\ln(x) - 1}{x - e}$$

1. The problem asks to complete the table by finding either the limit $\lim_{x \to c} f(x)$ or the function value $f(c)$ for each given function and point $c$. 2. Recall that if $f

1. **State the problem:** Find the first derivative $f'(x)$ and the second derivative $f''(x)$ of the function $$f(x) = \frac{x}{x^2 - 2}.$$\n\n2. **Recall the formula:** For a fun

1. **State the problem:** Differentiate the function $$F(t) = \frac{At^2}{Bt^5 + Ct^9}$$ with respect to $t$. 2. **Recall the quotient rule:** For a function $$\frac{u(t)}{v(t)}$$,

1. **State the problem:** Differentiate the function $$f(z) = (6 - e^z)(6z + e^z)$$ with respect to $$z$$. 2. **Recall the product rule:** For two functions $$u(z)$$ and $$v(z)$$,

1. **State the problem:** Find the derivative $f'(a)$ of the function $$f(x) = \frac{x^2}{x + 20}$$ at the point $a = 5$ using the definition of the derivative: $$f'(a) = \lim_{x \

1. **Problem statement:** Find the slope $m$ of the tangent to the curve $y = 7 + 5x^2 - 2x^3$ at $x = a$.

1. Consider the differential equation $$\frac{dy}{dx} = \frac{1}{2} x (y - 4)^3.$$ We are asked to find the particular solution passing through given points and analyze the slope f

1. **State the problem:** We are given the differential equation $$\frac{dy}{dx} = x - 1$$ and asked to analyze it through several parts including slope field, tangent line, and pa

1. **Problem:** Find the general solution to the differential equation $$2x(y+1) - y y' = 0$$. **Step 1:** Rewrite the equation to isolate $$y'$$.

1. **Problem:** Find the general solution to the differential equation $$2x(y + 1) - y y' = 0.$$ 2. **Step 1: Rewrite the equation**

1. **State the problem:** We need to evaluate the definite integral $$\int_0^{\frac{\sqrt{2}}{2}} \frac{-3x^2}{\sqrt{1 - x^2}} \, dx$$

1. **State the problem:** Evaluate the integral $$\int \frac{-9x^2}{\sqrt{4x - x^2}} \, dx.$$\n\n2. **Rewrite the expression under the square root:** Note that $$4x - x^2 = -(x^2 -

1. **State the problem:** Find the derivative of the function $f(x)$. 2. **Recall the derivative definition and rules:** The derivative of a function $f(x)$, denoted $f'(x)$ or $\f

1. **State the problem:** We have a particle moving along the y-axis with velocity function $v(t) = 12t^2 - 24t$ for $t \geq 0$. The initial position is $x(0) = 10$ cm. We need to

1. **Problem statement:** A particle moves along the x-axis with acceleration $a(t) = 12t + 6$. Given velocity at $t=3$ is 36 cm/sec and initial position $x(0) = 4$ cm, find the po

1. **Problem 1:** A particle moves along the x-axis with acceleration $a(t) = 12t + 6$, velocity at $t=3$ is 36 cm/sec, and initial position $x(0) = 4$ cm. Find the position when v