∫ calculus

1. **State the problem:** We need to evaluate the integral $$\int \frac{5}{13 - 6x + x^2} \, dx.$$\n\n2. **Rewrite the denominator:** The quadratic in the denominator is $$x^2 - 6x

1. Let's start by understanding the problem: you want to learn about functions and limits. 2. A function is a relation where each input has exactly one output. For example, $f(x) =

1. **Problem statement:** Find the limits: (i) $$\lim_{x \to \infty} \frac{x \sin x}{x^2 + 4}$$

1. **State the problem:** Find the second derivative of the function $$y = x^2 e^{2x}$$. 2. **Recall the product rule:** For two functions $u(x)$ and $v(x)$, the derivative is $$\f

1. **Problem Statement:** Test the continuity of the piecewise function

1. The problem is to explain Maclaurin's theorem, which is a special case of the Taylor series expansion of a function about zero. 2. Maclaurin's theorem states that any function $

1. **State the problem:** We want to find the Maclaurin series expansion of the function $$f(x) = e^{-x \sin m}$$ around $$x=0$$.

1. **Problem:** Find the derivative of the function $y = 3x^2 - 2$ at $x=1$. 2. **Formula:** The derivative of $y = ax^n$ is $y' = a n x^{n-1}$.

1. The question asks why we might not integrate with respect to $x$ first in a given problem. 2. When performing double integrals, the order of integration depends on the region of

1. **State the problem:** Find the limit $$\lim_{x \to 0^-} \frac{\sqrt{1+2x}-1}{3x}$$. 2. **Recall the formula and approach:** When direct substitution leads to an indeterminate f

1. **State the problem:** We want to find the limit as $x$ approaches positive infinity of the expression $$\sqrt{x^2 + \sqrt{x^2 + 1}} - x.$$\n\n2. **Recall the formula and approa

1. **State the problem:** Find the limit $$\lim_{x \to 2} \frac{x^3 - 8}{3 - \sqrt{2x + 5}}$$. 2. **Recognize the indeterminate form:** Substitute $x=2$ directly:

1. Masalah: Tentukan integral dari fungsi turunan $y' = x^2 + 6x - 8$. 2. Rumus yang digunakan: Integral dari turunan fungsi $y'$ adalah fungsi asli $y$, sehingga kita mencari $$y

1. **Problem statement:** Find the functions $y$ given their derivatives: 3. $\frac{dy}{dx} = \frac{2}{x^2 + 4}$

1. The problem is to find the function $y$ given its derivative $\frac{dy}{dx} = x^3 + 2x$. 2. We use the formula for antiderivatives: if $\frac{dy}{dx} = f(x)$, then $y = \int f(x

1. **State the problem:** We need to find the integral of the function $$\frac{x^2}{x^2 + 1}$$ with respect to $x$. 2. **Rewrite the integrand:** Notice that $$\frac{x^2}{x^2 + 1}

1. **State the problem:** We want to find the dimensions of the rectangle with the largest area that has its base on the x-axis and its other two vertices on the parabola $$y = -9x

1. **Problem Statement:** Find the value of $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}$ where $u = \frac{x^2 + y^2}{x + y}$.

1. **State the problem:** We want to maximize the function $$f(x,y) = 5y^2 - xy$$ subject to the constraint $$x = y^3 - y^2$$ and $$y \geq -2$$. 2. **Substitute the constraint into

1. **State the problem:** We need to evaluate the integral $$\int \frac{5}{|x| \sqrt{x^2 - 16}} \, dx.$$\n\n2. **Recall the formula and rules:** The integral involves an expression

1. The problem is to find an integral whose result is the inverse secant function, $\arcsec(x)$. 2. Recall that the derivative of $\arcsec(x)$ is given by the formula: