∫ calculus

1. **Evaluate** $$\int \frac{3y}{5y^2 + 1} dy$$ Step 1. Let us substitute $$u = 5y^2 + 1$$. Then, $$du = 10y dy$$ or $$dy = \frac{du}{10y}$$.

1. **Stating the problem:** We want to show that for a sequence $a_n > 0$ for all $n$, $\lim_{n\to\infty} a_n = \infty$ if and only if $\lim_{n\to\infty} \frac{1}{a_n} = 0$. 2. **R

1. Let's analyze the function A: $F(x) = x \sin x$. - Domain: $x$ can be any real number since $x$ and $\sin x$ are defined everywhere.

1. نبدأ بمسألة حساب نهاية الدالة عند $x \to 0$ للدالة $f(x) = \sin\left(\frac{3x^2}{5x}\right)$. \n2. يمكن تبسيط داخل دالة الجيب: $$\frac{3x^2}{5x} = \frac{3x}{5}.$$ \n3. إذن تصبح

1. **Problem statement:** Prove Lagrange's theorem (Mean Value Theorem), which states: If a function $f(x)$ is continuous on the closed interval $[a,b]$ and differentiable on the o

1. The problem: find the derivative of $\tan x$ with respect to $x$.\n\n2. Recall the definition: $\tan x = \frac{\sin x}{\cos x}$.\n\n3. Use the quotient rule: if $f(x) = \frac{g(

1. The problem asks us to differentiate the function $y=\tan x$. 2. Recall that the derivative of $\tan x$ with respect to $x$ is a standard derivative result from calculus.

1. The problem is to find the integral of the function $\sec^2(x)$. 2. Recall that the derivative of $\tan(x)$ is $\sec^2(x)$, so the antiderivative (integral) of $\sec^2(x)$ is $\

1. The problem is to find the integral of $\tan^2 x$ with respect to $x$. 2. Recall that $\tan^2 x = \sec^2 x - 1$ from the Pythagorean identity.

1. We are asked to find the integral of the function $\tan x$. 2. Recall that $\tan x = \frac{\sin x}{\cos x}$.

1. Stating the problem: Find the derivatives of the functions \(y = 3x^{-4} + 2x^{-3} + x^{-2} + x^{-1}\) and \(y = \cos^3(2x)\). 2. For \(y = 3x^{-4} + 2x^{-3} + x^{-2} + x^{-1}\)

1. Diberikan fungsi $f(x) = 6 - 2x$.\nTurunan dari $f(x)$ adalah turunan dari setiap suku.\nTurunan dari konstanta 6 adalah 0, dan turunan dari $-2x$ adalah $-2$.\nJadi, $$f'(x) =

1. The problem is to find the derivatives of each given function. 2. For $y=\sqrt{x}$, rewrite as $y=x^{1/2}$. Using the power rule, $\frac{dy}{dx}=\frac{1}{2}x^{-1/2}=\frac{1}{2\s

1. **State the problem:** We have the function $f(x) = \sqrt{4 + \ln(x)}$. We want to find its derivative $f'(x)$ and determine its domain. 2. **Find the derivative:**

1. **Problem:** Find the derivative and domain of the function $$f(x) = \sqrt{4 + \ln(x)}$$ and analyze related domain conditions for $$g(x) = \ln(x^2 - 12x)$$. 2. **Derivative of*

1. Problem: Calculate \(\lim_{\theta \to 0} \frac{\tan 5\theta}{\sin 2\theta}\). Step 1: Using the standard limit \(\lim_{x \to 0} \frac{\sin x}{x} = 1\) and \(\lim_{x \to 0} \frac

1. The problem is to perform the division involving one integral expression. 2. Let's consider an example: divide the integral $$\int_0^1 x^2\,dx$$ by a number, say 2.

1. Let's start by stating the problem: You want to learn how to divide integrals. 2. Remember that integrals themselves cannot be directly divided like regular numbers or expressio

1. The problem is to integrate the expression $$x^2 + \frac{1}{6}x - 9x^2$$ with respect to $$x$$. 2. First, combine like terms: $$x^2 - 9x^2 = -8x^2$$, so the expression inside th

1. **State the problem:** Calculate the integral $$\int \left(x^2 + \frac{1}{6}x - 9x^2\right) \, dx$$ and explain the steps clearly. 2. **Simplify the integrand:** Combine like te

1. Stated problem: Differentiate $T(z) = 4^z \log_4(z)$. 2. Recall the product rule for derivatives: if $T(z) = u(z)v(z)$, then $$T'(z) = u'(z)v(z) + u(z)v'(z)$$