∫ calculus

1. The problem asks us to evaluate the integral $$\int_1^4 \left(2e^{3x} + 5e^{\frac{x}{3}}\right) \, dx.$$\n\n2. We can split the integral into two parts: $$\int_1^4 2e^{3x} \, dx

1. **State the problem**: We need to evaluate the definite integral $$\int_1^3 \frac{4x^3 - 2x + 1}{x^2} \, dx.$$\n\n2. **Simplify the integrand** by dividing each term by $$x^2$$:

1. Stating the problem: We need to evaluate the definite integral $$\int_3^6 x \, dx$$. 2. Find the antiderivative of the integrand. The antiderivative of $$x$$ is $$\frac{x^2}{2}$

1. Let's start by understanding the **definition of a limit**: A limit $\,\lim_{x \to a} f(x) = L$ exists if as $x$ approaches a value $a$, the function $f(x)$ approaches a single

1. The problem asks to find $$\lim_{x \to 1} f(x)$$ given the graph. 2. We analyze the behavior of the function near $$x=1$$ from both sides.

1. **State the problem:** Find the limit $$\lim_{x \to 3} f(x)$$ given the graph with specified points and shape. 2. **Analyze the graph at $x=3$:** The graph has an open circle at

1. The problem asks us to find the limit of $f(x)$ as $x$ approaches 1, i.e., $\lim_{x \to 1} f(x)$.\n\n2. From the description, the graph of $f(x)$ has an open circle at $(1, 3)$

1. Stating the problem: We need to find the limit $$\lim_{x \to 2} (x^2 - 4)$$. 2. Understand the function: The expression inside the limit is a polynomial $$x^2 - 4$$, which is co

1. Problem a: Find the derivative of $$y=\frac{x \ln x}{1-x^2}$$ using the quotient rule. 2. Let $$u = x \ln x$$ and $$v = 1 - x^2$$. Then:

You mentioned that you will send 20 math questions on partial derivatives and want answers in abcd format without steps. Please send the questions so I can provide accurate answers

1. **Find the domain of the functions:** i) For $f(x) = \ln|x - 2|$:

1. The problem is to find the first and second derivatives of a given function, though the function was not explicitly provided in this message. 2. Assuming the function is known,

1. The problem is to find the 1st and 2nd derivatives of a function. 2. Since the function is not given, let's consider a general function $f(x)$.

1. The problem asks us to find the limit of the expression $$\frac{n+8}{n+4}$$ as $$n$$ approaches infinity.\n\n2. When $$n$$ becomes very large, the constants 8 and 4 become insig

1. Diketahui fungsi potong f(x) definisi: $$f(x) = \begin{cases} x^2 - a, & x < 2 \\ x + a, & x > 2 \end{cases}$$

1. The problem states the function $f(x) = |x + 1|$ and asks which limit statements about $f(x)$ are true. 2. Recall that the absolute value function $|y|$ returns the distance of

1. The problem states: Given the function $$f(x) = \frac{x^2 - 9}{x - 3}$$ and the limit condition $$\lim_{x \to 3} k f(x) = 18,$$ find the value of $k$. 2. First, factor the numer

1. State the problem: Compute $\lim_{x\to 4}\frac{\sqrt{2x+8}-4}{x-4}$.\n\n2. Observe that direct substitution gives $\frac{0}{0}$, an indeterminate form, so we must manipulate the

1. **State the problem:** Evaluate the limit \( \lim_{x \to 4} \left( \frac{\sqrt{2x+8}}{4} - 2 \right) \div \left( \sqrt{2x+8} - 4 \right) \).\n\n2. **Simplify the expression:**\n

1. Evaluate \(\int \frac{\sqrt{4x^2 - 1}}{x^2} \, dx\). Use substitution or integration by parts combined with algebraic manipulation to solve this integral.

1. State the problem: Find the limit $$\lim_{x\to 0} \frac{\ln(x+h) - \ln h}{x}$$ where $h>0$. 2. Recognize the expression as a difference quotient for the derivative of $\ln(x+h)$