∫ calculus

1. **State the problem:** We need to find the integral $$\int x^2 \sin(x) \, dx$$ using integration by parts. 2. **Recall the integration by parts formula:**

1. **State the problem:** We want to find the integral $$\int x^2 \sin(x) \, dx$$ using integration by parts. 2. **Recall the integration by parts formula:**

1. **State the problem:** We need to solve the differential equation $$y' = 6x^2 - 2at(2,22)$$. 2. **Interpret the equation:** The term $6x^2$ is straightforward, but $2at(2,22)$ i

1. **State the problem:** Find the limits of the given functions as $x$ approaches the specified values. 2. **Recall the limit evaluation rule:** If the function is continuous at t

1. **Problem Statement:** Find the limits of given functions as $x$ approaches specified values using tables of values and analyze right-hand and left-hand limits for parts (k), (m

1. **State the problem:** Find the limits of the following functions using tables of values: (k) $$\lim_{x \to 1} \sqrt{x^3 - 1}$$

1. **State the problem:** Find the definite integral $$\int_0^{\frac{\pi}{2}} \cos x \cdot e^{-\sin x} \, dx$$.

1. **Stating the problem:** We are asked to find the limits of the function $f(x)$ at various points: as $x$ approaches 0, 1 from the left, 1 from the right, exactly at 1, and as $

1. **Stating the problem:** We need to find the limits of the function $f(x)$ at various points based on the graph.

1. **State the problem:** We want to find the integral $$\int e^x \sin x \, dx$$. 2. **Formula and method:** We will use integration by parts twice. Recall integration by parts for

1. **Stating the problem:** We want to find the integral $$\int x \cos x \, dx$$. 2. **Formula and method:** We use integration by parts, which states:

1. **State the problem:** We need to evaluate the integral $$\int x^2 \sin(x^3) \, dx$$. 2. **Identify the method:** This integral suggests using substitution because the argument

1. **Problem:** Evaluate the integral $$\int \frac{dx}{x^2(x+1)}$$ **Step 1:** Use partial fraction decomposition:

1. **Problem Statement:** Estimate the area under the curve on the interval $[1,4]$ using the left-endpoint approximation with 3 rectangles. 2. **Formula and Explanation:** The lef

1. **Problem 7:** Identify the graph representing the function $$f(x) = x(x - 1)(x - 3)$$. 2. **Step 1:** Expand the function to understand its shape.

1. The problem asks on which interval the function $f(x) = 4 - x^2$ is increasing. 2. The problem asks when the function $f(x) = x(x - 1)^2$ has a local minimum.

1. **Problem Statement:** (a) Find the area common to the interiors of the cardioids $r = 1 + \cos \theta$ and $r = 1 - \cos \theta$.

1. **Problem Statement:** Show using the First Derivative Test that for the quadratic function $y = ax^2 + bx + c$ with $a \neq 0$, the graph has a relative maximum at the vertex i

1. **State the problem:** We want to evaluate the integral $$\int_{-t}^t x(t^2 - x^2) \, dx$$ using the power rule. 2. **Rewrite the integrand:** Distribute $x$ inside the parenthe

1. **State the problem:** We are given the differential equation $$\frac{dy}{dx} = (x-4)e^{-2y}$$ and asked to analyze or solve it. 2. **Identify the type of differential equation:

1. **State the problem:** We have a particle moving along the x-axis with acceleration function $a(t) = 2t + 4$. Given initial conditions: velocity at $t = -4$ is $v(-4) = 3$, and