∫ calculus

1. We are asked to find the integral $\int \sin 5x \, dx$. 2. The formula for integrating sine functions with a linear argument is:

1. **Problem:** Evaluate the integral $$\int \sqrt{x - 2} \, dx$$ 2. **Formula and substitution:** Use substitution $u = x - 2$, so $du = dx$. The integral becomes $$\int u^{1/2} \

1. **Stating the problem:** We are given the derivatives of three functions: $$f'(x) = -3x + 4$$

1. **State the problem:** Find the derivative of the function $y = \sin(\cot(5x))$ with respect to $x$. 2. **Recall the chain rule:** If $y = \sin(u)$ where $u = \cot(5x)$, then by

1. We are asked to evaluate the definite integral $$\int_0^4 (2e^x + 3 \sin x) \, dx$$. 2. The integral of a sum is the sum of the integrals, so we split it:

1. **Problem statement:** We are given the graph of $f'$, the derivative of $f$, for $x>0$. We need to determine which of the statements I, II, and III are true: I. $f(2) < f(5) <

1. **Problem statement:** Find the value of $x \geq 4$ that maximizes the area of the rectangle with vertices at $(4,0)$, $(x,0)$, $(4,-\frac{8}{4^3})$, and $(x,-\frac{8}{x^3})$ wh

1. **State the problem:** We need to find the left-hand limit of the function $f(t)$ as $t$ approaches 75 from the left, i.e., $\lim_{t \to 75^-} f(t)$. The table provides values o

1. **State the problem:** We are given a table of values for $t$ approaching some integer $a$ from the left side, and corresponding values of $f(t)$. We need to determine the integ

1. **Problem:** Find the domain and critical values of the function $$y = x^3 + 6x^2 + 12x - 1$$. 2. **Domain:** Since this is a polynomial, the domain is all real numbers, $$(-\in

1. **Problem:** Find the domain and critical values of the function $$y = x^3 + 6x^2 + 12x - 1$$. 2. **Domain:** Since this is a polynomial, the domain is all real numbers, $$\math

1. **State the problem:** We are given the differential equation $$\frac{dy}{dx} = \sin x$$ with the initial condition $$y(0) = 0$$. We want to find the function $$y(x)$$ and calcu

1. **State the problem:** We need to find the area of the shaded region bounded by the lines and curve: $$y=4$$ (horizontal line), $$x=\frac{y}{4}$$ (vertical line in terms of y),

1. **State the problem:** Find the derivative of the function $f(x) = x^3 \cdot \ln(x)$. 2. **Recall the product rule:** For two functions $u(x)$ and $v(x)$, the derivative of thei

1. **State the problem:** Find the limit $$\lim_{b \to 6} \frac{\frac{1}{b} - \frac{1}{6}}{b - 6}$$. 2. **Recall the formula:** This is a difference quotient resembling the definit

1. **State the problem:** Find the limit as $t$ approaches 2 of the expression $$\frac{t^2 - 4}{2t^2 - 13t + 18}$$. 2. **Recall the limit rule:** If direct substitution results in

1. The problem is to identify the behavior of the function based on the graph descriptions provided. 2. Let's analyze each graph description:

1. The problem states that the function $f$ is differentiable and increasing on the interval $0 \leq x \leq 6$, and that $f$ has exactly two points of inflection on this interval.

1. **Problem Statement:** Given two functions $F(x)$ and $G(x)$ with graphs provided, define $P(x) = F(x)G(x)$ and $Q(x) = \frac{F(x)}{G(x)}$. We need to find $P'(1)$, $Q'(1)$, $P'

1. **State the problem:** We need to find a function $f(x)$ such that: - $\lim_{x \to 2} f(x) = -\infty$

1. **State the problem:** We want to transform the integral $$\int \frac{x^6}{\sqrt{1+x^2}} \, dx$$ using the substitution $$x = \sec(\theta)$$ where $$0 \leq \theta < \frac{\pi}{2