∫ calculus

1. **Problem Statement:** Find the integral $$\int \frac{x+3}{(x-1)^2} \, dx$$ given that $$z = x - 1$$.

1. **State the problem:** We need to solve the integral $$\int \frac{1}{\ln x} \, d(\ln x)$$. 2. **Understand the integral:** The integral is with respect to $d(\ln x)$, which mean

1. **State the problem:** We are given the integral equation $$\int e^{2x} \left(\ln x + \frac{a}{x}\right) dx = \frac{1}{2} e^{2x} \ln x + c$$ and need to find the value of $a$. 2

1. **State the problem:** We want to find the values of $r$ and $h$ that maximize the total volume of the combined cylindrical cans, given the surface area constraint. 2. **Recall

1. **State the problem:** We want to solve the integral $$\int x (\sin^2(x) - \cos^2(x)) \, dx$$. 2. **Use trigonometric identities:** Recall that $$\sin^2(x) - \cos^2(x) = -\cos(2

1. We are asked to solve the integral $$\int x \sin(x) \cos(x) \, dx$$. 2. Use the double-angle identity for sine: $$\sin(2x) = 2 \sin(x) \cos(x)$$, so $$\sin(x) \cos(x) = \frac{\s

1. **State the problem:** We need to find the integral $$\int x^2 \cos(x) \, dx$$. 2. **Formula and method:** Use integration by parts, which states:

1. **State the problem:** We need to express the total volume $V$ of the combined cylindrical cans in terms of the radius $r$ of the smaller cylinder. 2. **Recall the volume formul

1. The problem is to understand why the differential $du$ equals $2x$ in the context of calculus. 2. Typically, $du$ represents the differential of a function $u$ with respect to $

1. **Problem statement:** (i) Find $$\int \sin^2 3x \, dx$$.

1. **Problem statement:** We have two cylinders, a small one with radius $r$ and height $h$, and a large one with radius $2r$ and height $2h$. The total surface area of both cylind

1. **Problem:** Calculate the indefinite integral $$\int (x + 5)(x - 9) \, dx$$. 2. **Formula and rules:** Use the distributive property to expand the integrand and then integrate

1. **Problem:** Calculate the indefinite integral \(\int \frac{5x + 3}{\sqrt{x^2 + 4x + 10}} \, dx\). 2. **Step 1: Simplify the expression under the square root.**

1. **Problem:** Calculate the indefinite integral $$\int \frac{5x+3}{\sqrt{x^2+4x+10}} \, dx$$. 2. **Step 1: Simplify the expression under the square root.**

1. **مسئله اول:** حد $$\lim_{x \to 3} \frac{x^2 - 9}{3x - 5 - 2}$$ را بیابید. فرمول: برای یافتن حد کسرها، ابتدا صورت و مخرج را ساده می‌کنیم و اگر مقدار در مخرج صفر شد، از روش‌های د

1. مسئله: حدهای زیر را به دست آورید. الف) $$\lim_{x \to 3} \frac{x^2 - 9}{\sqrt{3x - 5} - 2}$$

1. The problem is to find the Maclaurin series (Taylor series at $x=0$) expansion of the function $f(x) = e^{\cos x}$. 2. Recall the Maclaurin series formula for a function $f(x)$:

1. **State the problem:** Find the second derivative $f''(x)$ of the function $$f'(x) = (-3x^2 + 9x - 3)(x^2 - x + 1) - 3$$

1. The problem is to find the function $$F(x) = \int_{0}^{x} \lambda e^{-\lambda t} dt$$ where $$\lambda$$ is a constant. 2. We use the formula for the integral of an exponential f

1. **Stating the problem:** We want to find the function $$F(x) = \int_{0}^{x} \lambda e^{-\lambda t} dt$$ where $$\lambda > 0$$ is a constant. 2. **Formula and rules:** This is a

1. **State the problem:** Calculate the line integral \(\int_C (xy^2 + 1)\,dx + x(y^2 - 1)\,dy\) where \(C\) is the line segment from \((2,4)\) to \((4,6)\).