∫ calculus

1. **Problem Statement:** Solve the integral $\int x \, dx$. 2. **Formula Used:** The integral of $x$ with respect to $x$ is given by the power rule for integration:

1. **Problem statement:** Evaluate the integral $$\int \sqrt{\tan x} \, dx$$. 2. **Understanding the integral:** The integral involves the square root of the tangent function, whic

1. **State the problem:** We need to find the limit $$\lim_{x \to 5} \left(\frac{1}{5+x}\right)(10+2x)$$

1. **Problem Statement:** Find the minimum value of $x^2 + y^2 + z^2$ subject to the constraint $xyz = a^3$ using Lagrange multipliers. 2. **Formula and Concept:** To find extrema

1. **State the problem:** We are asked to find the left-hand limit, right-hand limit, two-sided limit, and the function value at $x = -1$ for the function $f(x)$ based on the given

1. **State the problem:** We need to find the limit of the function $f(x)$ as $x$ approaches $-1$ from the left, i.e., $\lim_{x \to -1^-} f(x)$. 2. **Understand the graph:** The gr

1. **Problem Statement:** Find the derivative of the function $f(x) = x + \frac{1}{x}$ using the definition of the derivative. 2. **Definition of Derivative:** The derivative of a

1. **Problem Statement:** We want to use Riemann sums to show that $$\int_a^b 3x^2 \, dx = b^3 - a^3.$$

1. **State the problem:** Find the points of intersection of the line $y=\frac{1}{2}x$ and the parabola $y=2x-\frac{1}{6}x^2$. 2. **Set the equations equal to find intersection poi

1. The problem is to find the limit $$\lim_{x \to 0} \frac{\sin x}{x}$$. 2. We use the standard limit rule: $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$.

1. **Problem statement:** (a)(i) Given the sequence $x_n = \frac{4n^2 + 3n + 99.5}{2n^2 + 5}$, find the limit $\ell$ as $n \to \infty$ assuming the sequence converges.

1. **State the problem:** Find all values of $x$ in the interval $0 < x < 2\pi$ for the function $f(x) = x - 2 \cos x$ where the slope of the tangent line is 2. 2. **Recall the for

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{x^3 - 3x^2 + x}{x^3 - 2x}$$. 2. **Recall the limit rules:** When direct substitution results in an indeterminate for

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{x^3 - 3x^2 + x}{x^3 - 2x}$$. 2. **Recall the limit rule:** If direct substitution leads to an indeterminate form lik

1. **State the problem:** Find the limit $$\lim_{x \to 2} \frac{3x^2 - 3x}{x - 0}$$. 2. **Rewrite the expression:** The limit is $$\lim_{x \to 2} \frac{3x^2 - 3x}{x}$$.

1. **State the problem:** We need to find the Jacobian of a transformation scaled by the determinant of the matrix \(\begin{bmatrix} 2 & 6 & 5 \end{bmatrix}\) and then evaluate the

1. **Problem statement:** We are given the function $z = \sqrt{3x + 4y}$ and asked to find the rate of change of $z$ at the point $(3,1)$ as $x$ changes while holding $y$ fixed. 2.

1. **State the problem:** Given the function $f(x) = (x^2 + 12)(9 - x^2)$, we need to find critical values, intervals of increase/decrease, local maxima/minima, and intervals of co

1. **Problem statement:** We have the function $$f(x) = x^6 (x - 6)^3$$ defined for $$x \in [-10, 11]$$. We need to find the intervals where $$f$$ is increasing, the region where $

1. **Problem Statement:** Find the points of intersection of the curves $y = x^3$ and $x = y^2$. 2. **Step 1: Express both equations clearly:**

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{\sec x}{\tan x}$$. 2. **Recall the formula:** To differentiate a quotient $$\frac{u}{v}$$, use the quot