∫ calculus

1. **State the problem:** We need to find the integral $$\int \sqrt{16x} \sin\left(1 + x^{32}\right) \, dx$$. 2. **Rewrite the integral:** Note that $$\sqrt{16x} = \sqrt{16} \sqrt{

1. **Problem statement:** We have a storage tank with 10,000 litres of chemical leaking at a rate given by the derivative of the amount spilled, $$f'(t) = 400e^{-0.01t}$$ where $$t

1. **State the problem:** We want to analyze the series $$\sum_{n=1}^\infty \left( \frac{n^2}{2^n} + \frac{1}{n^2} \right)$$ and determine its behavior or sum if possible. 2. **Rec

1. **State the problem:** We want to test the convergence of the series with general term $a_n = \frac{n! \times 3^n}{n^n}$ using the ratio test. 2. **Recall the ratio test formula

1. **State the problem:** We want to test the convergence of the series with general term $a_n = \frac{n! \times 3^n}{n^n}$ using the ratio test. 2. **Recall the ratio test formula

1. **State the problem:** We want to test the convergence of the series with general term $a_n = \frac{n! \times 3^n}{n^n}$ using the ratio test. 2. **Recall the ratio test formula

1. **State the problem:** We want to test the convergence of the series $$\sum_{n=1}^\infty \frac{1}{n^n}$$ using the ratio test. 2. **Recall the ratio test formula:** For a series

1. **State the problem:** Find the average value of the function $f(x) = \sqrt[3]{x}$ on the interval $[1,8]$. 2. **Formula for average value of a function:** The average value $f_

1. مسئله: تابع داده شده $y = -x^3 + m x^2 - 2 m x + 3$ است. می‌خواهیم تعداد مقادیر صحیح $m$ را بیابیم که رأس سهمی در ناحیه چهارم محورهای مختصات قرار دارد. 2. ابتدا باید بدانیم رأس

1. **State the problem:** Evaluate the definite integral $$\int_{\pi/6}^{\pi/4} \left(5 - 2 \sec z \tan z\right) \, dz.$$\n\n2. **Recall the integral formulas and rules:**\n- The i

1. **State the problem:** Evaluate the integral $$\int \frac{\csc^2(x)}{\cot(x)} \, dx$$. 2. **Recall the trigonometric identities:**

1. **State the problem:** We need to find the limit of the function $f(x) = x^3 - 1$ as $x$ approaches 1. 2. **Substitute the value:** Since $f(x)$ is a polynomial, it is continuou

1. **State the problem:** We have the function $$f(x) = \frac{x^2 - 15}{x - 4}$$ and need to find where it is increasing, decreasing, and its local extrema. 2. **Find the derivativ

1. The problem is to find the limit of a function or expression as the variable approaches a certain value. 2. Since the user only wrote "Lim" without specifying the function or th

1. The problem is to find the limit $$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$. 2. Notice that directly substituting $x = 3$ gives $$\frac{3^2 - 9}{3 - 3} = \frac{9 - 9}{0} = \frac{0

1. The problem is to find the limit $$\lim_{x \to 3} \frac{x^2 - 9x}{3}$$. 2. First, substitute $x = 3$ directly into the expression:

1. The problem is to find the limit $$\lim_{x \to 3} x^2 - 9x - 3$$. 2. First, substitute $x = 3$ directly into the expression:

1. **State the problem:** We have a storage tank with 10,000 litres of chemical leaking at a rate given by the derivative of the amount spilled, $$f'(t) = 400e^{-0.01t}$$ where $$t

1. **Problem statement:** Consider the function $f$ defined on $]0; +\infty[$ by $$f(x) = x - \frac{\ln x}{x}.$$

1. **State the problem:** We need to evaluate the definite integral $$\int_{-\pi}^{\frac{\pi}{2}} 3 \sin(x) + \cos(3x) \, dx.$$\n\n2. **Split the integral:** Use linearity of integ

1. **State the problem:** We are given the function $f(x) = \frac{e^x}{x^2}$ and need to find its derivative $f'(x)$. 2. **Identify the rule to use:** Since $f(x)$ is a quotient of