∫ calculus

1. **State the problem:** We want to find the integral $$I = \int x \tan^{-1}(x) \, dx.$$\n\n2. **Formula and method:** To solve this integral, we use integration by parts. Recall

1. **State the problem:** We need to evaluate the integral $$I = \int \frac{3x}{x^2 - 4x - 5} \, dx.$$\n\n2. **Identify the formula and approach:** The integral involves a rational

1. **State the problem:** We want to evaluate the integral $$I = \int \sin^3(4x) \cos^2(4x) \, dx.$$\n\n2. **Rewrite the integrand:** Use the identity $$\sin^3(4x) = \sin(4x) \cdot

1. **State the problem:** We want to evaluate the integral $$I = \int \frac{x \sin \sqrt{x^2 + 1}}{\sqrt{x^2 + 1}} \, dx.$$\n\n2. **Identify substitution:** Let $$u = \sqrt{x^2 + 1

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{e^{3x} - 1}{1 - 8^x}$$. 2. **Recall the formula and rules:** For limits of the form $$\frac{f(x)-f(a)}{g(x)-g(a)}$$

1. The problem is to find the Laplace transform of the function $f(t)$, denoted as $L\{f(t)\}$. 2. The Laplace transform is defined by the formula:

1. The problem is to analyze the function $f(t) = \frac{\sin(t)}{t}$ and understand its behavior. 2. This function is known as the sinc function (unnormalized). It is defined as $f

1. **Problem statement:** Find $a+b$ given that $$\lim_{x \to 2} \frac{(x^3 - 8)(x^2 + ax + b)}{(x^2 - 4)^2} = 6.$$ 2. **Rewrite and analyze the limit:** Note that $x^3 - 8 = (x-2)

1. **State the problem:** We need to find the integral with respect to $x$ of the function $$\frac{1}{\pi} \arccos \left(\frac{r-x}{r}\right).$$ 2. **Recall the formula and rules:*

1. **State the problem:** We want to evaluate the integral $$\int_0^r \frac{1}{\pi r^2} \arccos\left(\frac{r x - x^2 - \sqrt{\frac{2 r^3}{x} - \frac{3}{2} x r^2 - x^2 - \frac{2 r^3

1. **Stating the problem:** Evaluate the definite integral

1. **State the problem:** Evaluate the integral $$\int \frac{x^3 + 4}{x^2} \, dx.$$\n\n2. **Rewrite the integrand:** Simplify the expression inside the integral by dividing each te

1. נניח את הבעיה: לחשב את הגבול $$\lim_{x \to \infty} \frac{3x(x+1)^4}{(2x-1)^5}$$. 2. נשתמש בכלל הגבולות לפונקציות פולינומיות וחזקות: כאשר $x \to \infty$, הביטוי שגדל הכי מהר בדומ

1. **Problem:** Evaluate $\frac{dy}{dx}$ when $1 + xy^2 + x^2y = 0$. 2. **Formula and rules:** Use implicit differentiation. Differentiate both sides of the equation with respect t

1. **Stating the problem:** We are given the acceleration function $$a(t) = 0.012 + 0.008 \sin(0.5 t)$$ and need to find the velocity $$v(t) = \int_0^t a(s) \, ds$$ and the positio

1. **Problem Statement:** We have a spherical balloon with radius $r$ increasing at a constant rate of $\frac{dr}{dt} = 2$ cm/sec.

1. **State the problem:** The height $h$ of a cone is increasing at a rate of $\frac{dh}{dt} = 10$ cm/sec, and the radius $r$ is changing so that the volume $V$ remains constant. W

1. The problem is to understand the limit expression for the derivative: $$f'(x) = \lim_{h \to 0} \frac{12xh + 6h^2}{h}$$

1. Evaluate the integral $$\int e^x \cos 2x \, dx$$ Step 1. State the problem: We want to find $$\int e^x \cos 2x \, dx$$.

1. **State the problem:** We want to find the limit $$\lim_{x \to 0} \frac{\sin x - \sin x \cos x}{x^3}$$. 2. **Rewrite the expression:** Factor out $\sin x$ in the numerator:

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{\cos 2x - \cos x}{1 - \cos x}$$. 2. **Recall relevant formulas and rules:**