∫ calculus

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

1. Задлах асуудал: $y=\cos^2(3x)$ функцийн уламжлалыг олох. 2. $y=\cos^2(3x)$ гэдэг нь $y=(\cos(3x))^2$ гэсэн утгатай.

1. Evaluate $$\lim_{x \to -2} \sqrt{4x^2 - 2}$$ Substitute $x = -2$:

1. The problem is to find the limit as $n$ approaches infinity of the expression $$\frac{(n+2)(n+3)(n+4)}{n^3 - 2n + 5}.$$\n\n2. First, expand the numerator: $$(n+2)(n+3)(n+4).$$\n

Bereken de volgende limieten stap voor stap: 7. $$\lim_{x \to 2} \frac{x - 2}{\sqrt{x - 2}}$$

1. The problem asks for the derivative of the inverse cotangent function, $\cot^{-1} x$.\n\n2. Recall the derivative formula for the inverse cotangent function: $$\frac{d}{dx} \cot

1. **Problem:** Find the derivative of $f(x) = (3x^3 + 3x - 1)^{10}$. Step 1: Use the chain rule. Let $u = 3x^3 + 3x - 1$, then $f(x) = u^{10}$.

1. We are given the function $y = e^{x^3}$ and asked to find the second derivative $\frac{d^2y}{dx^2}$. 2. First, find the first derivative $\frac{dy}{dx}$ using the chain rule:

1. **State the problem:** We need to verify that the limits $$\lim_{x \to \infty} \left(\frac{2}{x}\right)^{e^{-x}} \quad \text{and} \quad \lim_{x \to 0} \left(\frac{1}{e^{\frac{1}