∫ calculus

1. **State the problem:** We want to compute the integral $$\int_1^\infty \frac{e^{-3x} - e^{-7x}}{x} \, dx.$$ This is a classic example where Frullani's integral formula can be ap

1. **State the problem:** We need to show that the curve given by the function $$y=\frac{2x+1}{3x+6}$$ is increasing for all $$x \neq -2$$. 2. **Find the derivative:** To determine

1. **State the problem:** We want to find the limit $$\lim_{n \to \infty} n \ln\left(1 - \frac{5}{n}\right).$$ 2. **Rewrite the expression inside the logarithm:** As $n$ becomes ve

1. **State the problem:** We need to find the limit as $n$ approaches infinity of the expression $$\frac{1 + (-1)^n}{2 + (-1)^n}.$$\n\n2. **Analyze the behavior of $(-1)^n$:** The

1. **Problem:** Find the derivative of the exponential function $y = e^{-3x}$. 2. **Recall the rule:** The derivative of $e^u$ with respect to $x$ is $$\frac{d}{dx} e^u = e^u \frac

1. The problem is to find the derivative $\frac{dy}{dx}$ given the parametric equations: $$x = t^3 + 1, \quad y = 4t^2 - 4t$$

1. **State the problem:** We want to analyze the function $$y = x \ln(4 + x^2) + 4 \arctan\left(\frac{x}{2}\right) - 2x$$ and understand its behavior. 2. **Rewrite the function:**

1. **State the problem:** Find the first derivative of the function $$y = x \ln(4 + x^2) + 4 \arctan\left(\frac{x}{2}\right) - 2x.$$\n\n2. **Differentiate each term separately:**\n

1. We want to evaluate the integral $$I = \int e^x \sin x \, dx.$$\n\n2. Use integration by parts. Let $$u = \sin x$$ and $$dv = e^x dx$$ so that $$du = \cos x dx$$ and $$v = e^x.$

1. **State the problem:** Find the first four non-zero terms of the Maclaurin series for the function $f(x) = \ln(1+x)$. 2. **Recall the Maclaurin series definition:** The Maclauri

1. **مسئله:** یافتن نقاط بحرانی تابع $$F(x) = x^{\frac{4}{5}} (x - 4)^2$$ است. 2. **تعریف نقاط بحرانی:** نقاطی هستند که مشتق اول تابع صفر یا تعریف نشده باشد.

1. مسئله: پیدا کردن نقاط بحرانی تابع داده شده است. 2. نقاط بحرانی نقاطی هستند که در آن‌ها مشتق اول تابع صفر می‌شود یا مشتق اول وجود ندارد.

1. **State the problem:** Evaluate the integral $$\frac{1}{2} \int_0^{\frac{\pi}{2}} \sin(x) (\sin(2x))^2 \, dx.$$\n\n2. **Rewrite the integrand:** Recall that $$\sin(2x) = 2 \sin(

1. Given $y = 4^{4x} + 6^x$, find $\frac{dy}{dx}$. Step 1: Recall the derivative of $a^{u}$ with respect to $x$ is $a^{u} \ln(a) \frac{du}{dx}$.

1. Differentiate $y = \sin^{-1}(3x)$. Step 1: Recall the derivative formula for $y = \sin^{-1}(u)$ is $\frac{dy}{dx} = \frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx}$.

1. Differentiate $y = \sin^{-1}(3x)$. Using the chain rule, the derivative of $\sin^{-1}(u)$ is $\frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx}$.

1. Problem: Find the radius of curvature for $y=2x^2$ at the point $(1,2)$. Step 1: Compute the first derivative $y' = \frac{dy}{dx} = 4x$.

1. The problem is to evaluate the definite integral $$\int_5^8 \sin(x)\,dx$$. 2. Recall that the antiderivative of $\sin(x)$ is $-\cos(x)$.

1. The problem is to find the integral of $\cos^2 \theta$ with respect to $\theta$. 2. Use the trigonometric identity to simplify the integrand:

1. **Stating the problem:** We analyze the cubic function $$y = x(x-6)^2$$ which has critical points at $$x=2$$ and $$x=6$$ with given values $$f(2)=8$$ and $$f(6)=0$$. 2. **Find t

1. Таны асуусан "dy/dx" гэдэг нь математикт дифференциалчлалын тэмдэглэгээ юм. 2. Энэ нь "y" хувьсагчийн "x" хувьсагчийн хувьд авсан дериватив буюу хамааралтай хувьсагчийн өөрчлөлт