∫ calculus

1. **Menentukan luas daerah yang dibatasi oleh kurva $y = \sin x$ dan $y = \sin 2x$ antara $x=0$ dan $x=\frac{\pi}{3}$** Luas daerah antara dua kurva $y=f(x)$ dan $y=g(x)$ pada int

1. **State the problem:** We need to find the derivative of the function $$F(x) = (6x^2 + 2x)(3x + 2)^2$$ with respect to $$x$$. 2. **Recall the product rule:** For two functions $

1. The problem is to understand and explain Leibniz Theorem. 2. Leibniz Theorem is a rule used in calculus for finding the $n^{th}$ derivative of a product of two functions.

1. **State the problem:** Differentiate the function $$f(t) = (5t + e^t)(3 - \sqrt{t})$$ with respect to $$t$$. 2. **Formula used:** Use the product rule for differentiation: $$\fr

1. **State the problem:** Differentiate the function $$f(x) = \frac{-10x+3}{-8x-3}$$ with respect to $$x$$. 2. **Formula used:** For a function $$f(x) = \frac{u(x)}{v(x)}$$, the de

1. **Problem Statement:** Evaluate the limit $$\lim_{n \to \infty} \frac{10n^2 + 4n + 1}{7n^4 + 5n}$$. 2. **Formula and Rules:** When evaluating limits at infinity for rational fun

1. **State the problem:** Evaluate the improper integral $$\int_1^\infty \frac{dx}{x\sqrt{x}}$$ and determine if it converges. 2. **Rewrite the integrand:** Note that $$\frac{1}{x\

1. The problem is to find the derivative of the function $y = e^{\frac{1}{x}}$. 2. We use the chain rule for differentiation: if $y = e^u$, then $y' = e^u \cdot u'$, where $u = \fr

1. **State the problem:** Given the functions and values: $f(1)=3$, $f'(1)=4$, $g'(1)=1$, $g(2)=5$, and the function definition $f(x) = f(2x+1)g(1 - x^2)$, find $f'(0)$.

1. **Problem:** Evaluate the integral $$\int \frac{2e^{2}\sinh x}{e^{2-x} + e^{2+x}} \, dx$$. 2. **Recall the definitions and formulas:**

1. **Problem Statement:** We are analyzing different types of discontinuities in functions at specific points: at $x=-1$, $x=1$, $x=2$, and $x=0$. 2. **Key Concepts:**

1. **State the problem:** We need to find the critical points of the function $$g(t) = t^2 \sqrt[3]{2t} - 5$$. Critical points occur where the derivative $$g'(t)$$ is zero or undef

1. **State the problem:** Find the limit $$\lim_{x \to +\infty} |x| \times \sqrt{x} - 2$$. 2. **Understand the expression:** Since $x \to +\infty$, $|x| = x$ because $x$ is positiv

1. **Problem Statement:** We have a smooth curve $f(x)$ defined on $(-\infty, \infty)$ with:

1. مسئلہ بیان کریں: ہمیں دی گئی پہلی اور دوسری مشتق کے سائن چارٹ کی مدد سے فنکشن $f$ کے بڑھنے، گھٹنے، محدب اور مقعر ہونے کے وقفے اور انفلکشن پوائنٹس تلاش کرنے ہیں۔ 2. اصول اور فارم

1. **State the problem:** Evaluate the limit $$\lim_{x \to \infty} \left( \frac{x^3}{3} \cdot 40 \cdot \frac{\pi}{4} - \arctan(x) + \frac{1}{3} \left( \frac{x^2}{2} + 0.5 \log(1 +

1. **State the problem:** Evaluate the integral $$\int_0^\infty r^2 \left(\frac{\pi}{4} - \arctan r\right) dr.$$\n\n2. **Recall relevant formulas and properties:** The integral inv

1. **Problem statement:** Given the function $y = \tan^{-1}\left(\frac{x}{a}\right)$, find the $n$th derivative $y^{(n)}$ with respect to $x$. 2. **Recall the formula for the first

1. **State the problem:** Differentiate the function $$f(x) = \frac{4}{(5-x)(x-1)}$$ with respect to $$x$$. 2. **Rewrite the function:** To differentiate, it's easier to write $$f(

1. **State the problem:** We need to evaluate the integral $$\int x \ln(x^2) \, dx$$. 2. **Recall the formula and rules:** Use integration by parts formula:

1. **State the problem:** Analyze the function $$f(x) = e^{2x} - 3e^x - 2x$$ including its limits as $$x \to \pm \infty$$, derivative, critical points, variation, and intersections