∫ calculus

1. Diberikan fungsi $y = e^{2x} \sin 3x$. Kita diminta mencari turunan ketiga dari fungsi ini. 2. Gunakan aturan turunan produk: jika $y = u \cdot v$, maka $y' = u'v + uv'$.

1. We are asked to evaluate the definite integral $$\int_0^1 x^2 \sqrt{1 + 4x^2} \, dx.$$\n\n2. To solve this integral, we use substitution. Let $$u = 1 + 4x^2.$$ Then, $$\frac{du}

1. **Problem statement:** Expand the function $f(x,y) = e^{x+y}$ using Taylor series about the point $(0,0)$ up to second degree terms. 2. **Formula:** The Taylor series expansion

1. Masalah: Hitung luas di bawah kurva menggunakan prosedur 3 langkah Riemann untuk grafik a dengan fungsi $f(x) = -x^2 + 4x$ pada interval $[1,4.5]$ dengan partisi $1, 2, 2.5, 3,

1. **Problem Statement:** Calculate the definite integral $$\int_\alpha^\beta \sqrt{(x-\alpha)(\beta - x)}\, dx$$ where $\alpha < \beta$. 2. **Formula and Approach:** This integral

1. Diberikan fungsi $f(x) = \ln(x^2 + 3x)$. Kita diminta mencari turunan fungsi tersebut, yaitu $f'(x)$.\n\n2. Gunakan aturan rantai untuk turunan fungsi logaritma natural: jika $f

1. **State the problem:** Evaluate the limit $$\lim_{x \to 0} \frac{\sin 3x}{x}$$. 2. **Recall the standard limit:** We know that $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$.

1. **State the problem:** We need to evaluate the definite integral $$\int_0^1 7x \cdot 7^{x^2} \, dx$$. 2. **Rewrite the integral:** Notice that $$7^{x^2} = e^{x^2 \ln 7}$$, so th

1. **Problem Statement:** Find the derivative $\frac{dy}{dx}$ using implicit differentiation for an equation involving both $x$ and $y$. 2. **Formula and Rules:** When differentiat

1. مسئله: حد چپ و راست تابع $\sin x$ در نقطه $\pi$ را پیدا کنیم. 2. فرمول و نکات مهم: تابع $\sin x$ یک تابع پیوسته و متناوب است که در هر نقطه مقدارش برابر با مقدار تابع در آن نقطه

1. **Stating the problem:** We need to determine the limit $$\lim_{x \to 2^-} \frac{f(x)}{x-2}$$ where the function $f(x)$ has vertical asymptotes at $x=0$ and $x=1$, and the behav

1. **State the problem:** We have a curve $C$ with equation $y = f(x)$ where $$f(x) = 2x^3 - kx^2 + 14x + 24,$$ and $k$ is a constant.

1. **State the problem:** Evaluate the integral $$\int 5^{\frac{x}{2}} \, dx$$. 2. **Recall the formula:** The integral of an exponential function with base $a$ is given by $$\int

1. **State the problem:** We need to evaluate the integral $$\int \frac{\ln x}{2x^3} \, dx$$. 2. **Rewrite the integral:** Express the integral as $$\int \frac{\ln x}{2x^3} \, dx =

1. Muammo: Quyidagi limitni hisoblang: $$\lim_{n \to \infty} \frac{n^n}{\left[(n+1)!\right]^2}$$

1. **State the problem:** Find the limit as $n$ approaches infinity of the sequence $$a_n = 1 + \sqrt{2} + \sqrt{3} + \cdots + \sqrt{n}.$$\n\n2. **Understanding the problem:** We w

1. The problem asks to evaluate the infinite series $$\sum_{n=1}^\infty \frac{n!}{n^{n-1}}$$ and check its convergence. 2. The general term of the series is $$a_n = \frac{n!}{n^{n-

1. **State the problem:** Determine the type of discontinuity of the function $$h(x) = -3^{\frac{1}{x}+1}$$ at $$x = -1$$. 2. **Recall the types of discontinuities:**

1. **State the problem:** Evaluate the limit $$\lim_{x \to 0} \frac{x - 1}{\sqrt{x + 3} - 2}$$. 2. **Recall the formula and approach:** When direct substitution leads to an indeter

1. **State the problem:** Find the maximum and minimum values of the function $f(x,y) = x^2 + y^2$ subject to the constraint $x^2 + y^2 = 1$ using Lagrange multipliers. 2. **Recall

1. **State the problem:** We have a piecewise function $$h(z) = \begin{cases} 6z & \text{if } z \leq -4 \\ 1 - 9z & \text{if } z > -4 \end{cases}$$