∫ calculus

1. The problem is to find the second derivative $f''(x)$ of the function $f(x) = 7x^3 - 6x^5$. 2. Recall the power rule for derivatives: if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

1. **State the problem:** Differentiate the function $v(t) = V(1 - e^{-t/T})$ with respect to $t$. 2. **Recall the differentiation rules:**

1. The problem is to find the integral of $\cos 3x$ with respect to $x$, i.e., compute $\int \cos 3x \, dx$. 2. Recall the formula for integrating cosine of a linear function: $\in

1. **State the problem:** We need to find the indefinite integral $$\int (6e^{2x} + 6x) \, dx$$. 2. **Recall the integration rules:**

1. **State the problem:** Differentiate the function $$f(x) = \frac{(x^2 + 1)^3}{x}$$ with respect to $x$. 2. **Rewrite the function:** To differentiate more easily, express $f(x)$

1. The problem is to differentiate a function using the product rule. 2. The product rule states that if you have two functions $u(x)$ and $v(x)$, then the derivative of their prod

1. The original function is $v = \sqrt{4x^2 - 1}$. 2. You found the derivative $v' = 4x(4x^2 - 1)^{-1/2}$.

1. The problem is to find the derivative $v'$ of the function $v = \sqrt{4x^2 - 1}$. 2. We use the chain rule for derivatives: if $v = (f(x))^{1/2}$, then $v' = \frac{1}{2}(f(x))^{

1. **State the problem:** Differentiate the function $$y = e^{2x} \times \sqrt{4x^2 - 1}$$ using the chain rule. 2. **Recall the product rule and chain rule:**

1. Problem from Seite 51 Nummer 8: Given the function $f(x) = -x^3 + 3x^2$, we are to split the area enclosed by the graph of $f$ and the x-axis by a vertical line so that the two

1. We are asked to evaluate the definite integral $$\int_1^{10} \frac{2624}{x^2 + x + 1} \, dx$$. 2. The integral involves a rational function with a quadratic denominator. To solv

1. **State the problem:** We need to evaluate the definite integral $$\int_1^{10} \frac{2624}{x^2 + x + 1} \, dx$$. 2. **Recall the formula and approach:** The integral of the form

1. **State the problem:** Evaluate the double integral $$\int_0^\infty \int_0^{\frac{\pi}{2}} \frac{x \sin \theta \ln(1 + x^2 \cos^2 \theta)}{(1 + x^2 \sin^2 \theta)^{3/2}} \, d\th

1. **State the problem:** Calculate the integral $$\int_0^{\ln 2} 2\pi \left( \frac{e^y + e^{-y}}{2} \right) \sqrt{1 + \left( \frac{e^y - e^{-y}}{2} \right)^2} \, dy$$.

1. **State the problem:** Evaluate the integral $$\int \ln\left(2\pi e^{y}+e^{-y}\right) \frac{\left(\frac{e^{y}+e^{-y}}{2}\right) \sqrt{1+\left(\frac{e^{y}-e^{-y}}{2}\right)^2}}{2

1. مسئله: مرکز سطح قسمت هاشور خورده زیر منحنی تابع $y = x^2 - x$ و بالای خط $y=2$ را بیابید. 2. ابتدا باید محدوده‌ی انتگرال‌گیری را مشخص کنیم. این محدوده نقاط تقاطع منحنی و خط $y=2

1. **State the problem:** We are given graphs of two functions $f(x)$ and $g(x)$ and asked to evaluate limits and function values involving $f(x)+g(x)$ at specific points. 2. **Rec

1. **State the problem:** We are asked to evaluate limits and function values involving the sum $f(x) + g(x)$ at points $x=1$ and $x=2$ using the given graphs. 2. **Recall limit an

1. **State the problem:** Determine the convergence or divergence of the series using the Comparison Test. 2. **Recall the Comparison Test:**

1. **State the problem:** Find the partial derivative of the function $$f(x,y) = e^{xy} \cos(x) \sin(y)$$ with respect to $$y$$, denoted as $$f_y(x,y)$$. 2. **Recall the formula an

1. **Problem statement:** Given the function $$z = x^2 + y^2$$, find the sign of the partial derivatives $$f_x(1,1)$$, $$f_x(-1,1)$$, $$f_y(1,1)$$, and $$f_x(0,0)$$ using the graph