∫ calculus

1. \textbf{المسألة:} لدينا الدالة $$f(x) = \frac{x^2 + x + 1}{x + 1}$$ ونريد حساب الحدود التالية: $$\lim_{x \to -\infty} f(x), \quad \lim_{x \to 0^-} f(x), \quad \lim_{x \to 0^+} f

1. **State the problem:** Calculate the definite integral $$5 \int_1^5 \frac{2x^2}{3} \, dx$$. 2. **Formula and rules:** The integral of a function $f(x)$ over $[a,b]$ is given by

1. **State the problem:** Estimate the area under the curve of the function $f(x) = x^2$ from $x=0$ to $x=1$ using the midpoint rule with first 2 rectangles, then 4 rectangles. 2.

1. **State the problem:** Evaluate the integral $$\int 6 \sqrt[3]{x} \left( \frac{x^3}{3} + \frac{7}{2} x^{-\frac{1}{3}} \right) dx.$$\n\n2. **Rewrite the integral:** Recall that $

1. The problem is to evaluate the integral $$\int x^3 \sin x + y^n \, du$$. 2. Since the integral is with respect to $u$, and the integrand contains $x^3 \sin x + y^n$ which are co

1. **Problem statement:** Given the function $y = f(x) = ax^3 + bx^2 + cx + d$ with the derivative sign table and function behavior: - $f'(x)$ changes sign: positive on $(-\infty,

1. **Problem statement:** Differentiate the function $f(x) = x^{100}$ using the binomial formula. 2. **Recall the binomial formula:** The binomial theorem states that for any integ

1. **Problem Statement:** Given that $$\lim_{x \to 3} f(x) = 7$$, determine which of the following statements must be true: I. $$f$$ is continuous at $$x=3$$

1. The problem asks to find the acceleration of the particle at time $t=\pi$ given the position function $x(t) = \sin(2t) - \cos(3t)$ for $t \geq 0$. 2. Acceleration is the second

1. **Problem statement:** Find the area under the curve $y=4-x^2$ on the interval $[-2,2]$ using rectangles. 2. **Formula and explanation:** The area under a curve can be approxima

1. **State the problem:** Find the maximum and minimum values of the function $$f(x) = 3x^4 - 2x^3 - 6x^2 + 6x + 1$$. 2. **Formula and rules:** To find maxima and minima, we first

1. **Problem statement:** Evaluate the limits (i) $$\lim_{x \to 0} \left( \frac{1}{\sin x} - \frac{1}{x} \right)$$

1. **Problem 1: Define and find values of the function** Given the function:

1. Problem (a): A balloon is rising vertically at 3 m/s. A boy cycles beneath it at 5 m/s. When the balloon is 40 m high, find the rate of change of the distance between them. 2. L

1. **State the problem:** Find the area of the shaded region bounded by the curve, the line $y=2$, and the vertical line $x=\pi$. 2. **Identify the boundaries:** The region is boun

1. **Problem:** Find the limit $\lim_{x \to 0} x \cot x$. 2. **Formula and rules:** Recall that $\cot x = \frac{\cos x}{\sin x}$ and near zero, $\sin x \approx x$ and $\cos x \appr

1. **State the problem:** We have the curve $$y = 3 + 2x - x^2$$ and a point $$A$$ on the curve at $$x=1.5$$. The tangent at $$A$$ meets the x-axis at $$B$$, and the curve meets th

1. **Problem statement:** Find the following limits:

1. **State the problem:** We need to find the indefinite integral $$\int (1 - 2x^2)^5 \, dx$$. 2. **Formula and substitution:** Use substitution for integrals of the form $$\int (f

1. **Problem statement:** Evaluate the integral $$I = \int \cot^3 x \csc^4 x \, dx.$$\n\n2. **Recall definitions and identities:** \n- $\cot x = \frac{\cos x}{\sin x}$\n- $\csc x =

1. **State the problem:** We need to evaluate the integral $$I = \int \frac{dx}{\sqrt{2x^2 + 3x + 4}}.$$\n\n2. **Identify the integral type:** This is an integral of the form $$\in