∫ calculus

1. Problem: Find the limit of the function $F(x) = 3$ as $x$ approaches $-1$. 2. Since $F(x)$ is a constant function, its value does not depend on $x$. Therefore, for any $x$, incl

1. **Problem 1:** Given $f(x)=\frac{(1+2x)^2}{1 - x^2}$, find the first 4 terms in the power series expansion and state when the expansion is valid. 2. **Step 1:** Expand numerator

1. **State the problem:** We want to find the integral $$I = \int x^2 e^x \sin x \, dx$$. 2. **Integration by parts formula:** $$\int u \, dv = uv - \int v \, du$$.

1. The problem asks: Which of the following functions is not differentiable? 2. The functions given are:

1. The problem states that: Given $y = (x^3 + 2)^7$, prove that $\frac{dy}{dx} = 21x^2 y$. 2. Start by differentiating $y$ using the chain rule. Let $u = x^3 + 2$. Then $y = u^7$.

1. The problem states: Given $y=(x^3+2)^7$, prove that $\frac{dy}{dx} = 21x^2y/(x^3+2)$. 2. Start by differentiating $y$ using the chain rule. Let $u = x^3 + 2$, so $y = u^7$.

1. The problem asks to prove that if $y=(x^3 + 2)^7$, then $$\frac{dy}{dx} = 21 x^2 y.$$\n\n2. Start with the given function $$y = (x^3 + 2)^7.$$\n\n3. To find $$\frac{dy}{dx}$$, w

1. Diberikan integral tak tentu $$\int \frac{-x^3}{(x-2)^5} \, dx$$. 2. Untuk menyelesaikan, kita coba substitusi: misalkan $$u = x-2$$ maka $$x = u+2$$ dan $$dx = du$$.

1. The problem is to find the integral $$\int \frac{-x}{(x-2)^{3/2}} \, dx$$

1. Problem 1: The graph is given as a function f with a specific shape from x=0 to x=1. (a) To find where f is increasing, look for intervals where the graph moves upwards as x inc

1. **State the problem:** Differentiate the function $$f(x) = \frac{(3 \cdot \sqrt[3]{x} + 2)^2}{2x^5}.$$

1. **State the problem:** We need to find the derivative $f'(x)$ of the function $$f(x) = \frac{1 + x^8}{5x}$$. 2. **Rewrite the function:** To differentiate easily, rewrite the fu

1. We begin by stating the problem: we need to find the derivative of the function $f(x)$ with respect to $x$, denoted $f'(x)$. 2. Since the function $f(x)$ is not explicitly given

1. The problem is to find the indefinite integral $\int \sin 2x \, dx$. 2. Recall the integral formula: $\int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + C$ where $a$ is a constant.

1. Stating the problem: Given $y = 3^{2 \log_3 x}$, find the second derivative $\frac{d^2 y}{dx^2}$.\n\n2. Simplify the expression using properties of logarithms and exponents. Rec

1. We are asked to evaluate the limit as $n \to 0$ of the expression $$\int_0^\pi (\sin \theta - \sin \theta) \lim_{n\to 0} \frac{\sqrt{n+\theta} - \sqrt{n+\theta}}{\theta} d\theta

1. Problem 9: Find the functions $f(x)$ given their derivatives $f'(x)$. 2. To find $f(x)$, integrate each derivative function with respect to $x$. Remember to add a constant of in

1. **Problem Statement:** Given the function $$g(x) = \frac{x^2 + 9}{|x - 3|}$$ (a) Determine the domain of $$g(x)$$.

1. **State the problem:** Determine if the limit exists:

1. The problem is to compute the expression $l[e^{-2t}\sinh 4t]$. 2. Recall that the Laplace transform of $\sinh(at)$ is $\frac{a}{s^2 - a^2}$ for $s > |a|$.

1. We evaluate each limit step-by-step. 2. a) $$\lim_{x\to4} \frac{x^2 - 16}{x - 4}$$ Factor numerator: $$x^2 - 16 = (x-4)(x+4)$$ Cancel $$x-4$$ term: