∫ calculus

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

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{\sqrt{x^2 + 100} - 10}{x^2}.$$\n\n2. **Recall the formula and approach:** When direct substitution leads to an indet

1. **State the problem:** We need to find the integral $$\int \sin(2x) e^{2x} \, dx$$. 2. **Formula and method:** This is an integral of the form $$\int e^{ax} \sin(bx) \, dx$$. Th

1. **State the problem:** Evaluate the integral $$\int \sin(5x) \cdot \cos(2x) \, dx.$$\n\n2. **Use product-to-sum formula:** Recall the identity $$\sin A \cos B = \frac{1}{2} [\si

1. **State the problem:** We need to find the local maxima and minima of the function $$f(x) = x^3 - 6x^2 + 9x + 1$$ using the second derivative test. 2. **Find the first derivativ

1. **Stating the problem:** We are given a graph representing the detection function $f(x)$ of a Tesla Optimus robot's LIDAR sensor. The graph oscillates with diminishing amplitude

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

1. **State the problem:** Find the limit $$\lim_{t \to 0} \tan t$$ using L'Hospital's rule. 2. **Recall the formula and rule:** L'Hospital's rule applies to limits of the form $$\f

1. Problem: Find the first and second derivatives $f'(x)$ and $f''(x)$ for the function $f(x) = 2^{2x+1}$. 2. Formula and rules:

1. **Problem:** Evaluate the integral $$\int x^3 \, dx$$. 2. **Formula:** The integral of $$x^n$$ with respect to $$x$$ is given by:

1. **Problem Statement:** We are given the expression:

1. **Problem:** Determine whether the function $f(x) = 3x^2 - 2x + 5$ is continuous at $x = 2$. 2. **Recall the definition of continuity at a point:** A function $f$ is continuous

1. **State the problem:** We are given a curve $y = \frac{9}{\sqrt{5x+4}}$ and a line $y = 6 - 3x$. They intersect at point $P$ where the $y$-coordinate is 3. We need to find the a

1. **State the problem:** Given $y = \log(2t)$ and $x = e^{-2t}$, prove that $$\frac{d^2y}{dx^2}\bigg|_{t=1} = \frac{e^4}{4}.$$\n\n2. **Recall the chain rule for derivatives:** To

1. The problem states that the derivative of the function $f$ at $x = -8.9$ is $f'(-8.9) = 4.53073$. 2. This means the slope of the tangent line to the curve of $f$ at the point wh

1. The problem asks to find intervals where the function $f(x)$ is concave up on the open interval $(-9,9)$. 2. Recall that concavity of $f$ is determined by the sign of the second

1. **State the problem:** Find the limit as $x \to \infty$ of $$8e^{\sqrt{\frac{x}{x+1}}} \left(\frac{x-1}{x}\right)! - \left(8x^2 - 4x\ln x - \ln 2x - \frac{4x + 2\ln x}{\ln 2\pi}