∫ calculus

1. The problem is to understand how to compute an integral, which is the process of finding the area under a curve or the antiderivative of a function. 2. The basic formula for an

1. **State the problem:** Find the integral of the function $5x^2 - 2x$ with respect to $x$. 2. **Recall the formula:** The integral of $x^n$ with respect to $x$ is given by $$\int

1. The problem is to evaluate the integral using integration by parts formula: $$\int u\,dv = uv - \int v\,du$$

1. **State the problem:** Find the general solution of the differential equation $$y' = \ln x$$. 2. **Recall the formula:** Since $$y' = \frac{dy}{dx}$$, the equation means $$\frac

1. Problem statement: Find the stationary point(s) of the curve $y=(3x^2+8)^{5/3}$ and determine their nature.

1. **Problem:** Find the derivative $f'(x)$ of the function $f(x) = x^6$. 2. **Formula:** Use the power rule for derivatives: $$\frac{d}{dx} x^n = n x^{n-1}$$ where $n$ is any real

1. **Problem statement:** Solve the integral of the ellipse equation $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ with respect to $x$ or $y$ within given limits. 2. **Understanding th

1. **State the problem:** We want to evaluate the double integral

1. Problem statement: Find the equation of the normal to the curve $\ln(2x^2 - 7)$ at the point where the curve crosses the positive x-axis. 2. Find the x-intercept by setting the

1. **State the problem:** Optimization involves finding the maximum or minimum value of a function, often subject to certain constraints. 2. **Identify the function to optimize:**

1. Diberikan fungsi: $$y = \sqrt{(3x^2 + 2)^5} \cdot \sin(4x - 1)$$ Kita diminta mencari turunan pertama $$\frac{dy}{dx}$$.

1. **State the problem:** Find the limit as $t \to +\infty$ of $$\frac{2}{t^2} - 4t$$ using limit theorems. 2. **Recall theorem 13 (Sum/Difference of limits):** If $\lim_{x \to a}

1. **Problem Statement:** Compute the integral $$\int \frac{4x^3 + 2x}{(x^4 + x^2 + 5)^5} \, dx$$.

1. **Problem:** Determine $f'(x)$ given that $f(x) = 4x^3$ from first principles. 2. **Formula:** The derivative from first principles is given by:

1. **State the problem:** Find the derivative of the function $f(x) = \ln x$. 2. **Recall the formula:** The derivative of the natural logarithm function $\ln x$ with respect to $x

1. **State the problem:** Find the derivative of the function $f(x) = e^x$. 2. **Formula used:** The derivative of the exponential function $e^x$ with respect to $x$ is given by:

1. **Problem 1: One-sided and two-sided limits from the graph** Given the graph of $y=f(x)$ with a horizontal asymptote at $y=-2$ and a vertical asymptote at $x=2$, we find the lim

1. **Problem:** Find the one-sided and two-sided limits of $f(x)$ at given points using the graph. 2. **Recall:** The one-sided limits are $\lim_{x \to a^-} f(x)$ (from the left) a

1. **State the problem:** We need to find and sketch the derivative of the function $$f(x) = \ln(\tan x)$$. 2. **Recall the formula for the derivative of a composite function:** If

1. **Problem:** Find the approximate change in volume $V$ of a cube of side $n$ meters caused by increasing the side by 1%. 2. **Formula and explanation:** The volume of a cube is

1. The problem states: If $$\frac{d}{dx} \sin(5x + b) = a \cos(5x + 4)$$, find the value of $$(a + b)$$. 2. Recall the derivative formula for sine: $$\frac{d}{dx} \sin(u) = \cos(u)