∫ calculus

1. The problem is to find the integral of $\sin(2x + y)$ with respect to $x$. 2. The integral formula for $\sin(ax + b)$ with respect to $x$ is:

1. The problem is to understand and apply the method of $u$-substitution in integration. 2. $u$-substitution is used to simplify integrals by substituting a part of the integral wi

1. **State the problem:** We need to evaluate the triple integral $$\int_0^{\pi/2} \int_0^y \int_0^x \cos(x + y + z) \, dz \, dx \, dy.$$\n\n2. **Understand the integral:** The int

1. **Problem statement:** Find and classify all stationary points of the function $$f(x) = x^3 - 6x^2 + 9x + 2$$. 2. **Find stationary points:** Stationary points occur where the f

1. **State the problem:** We need to find the equation of a curve such that at every point $(x,y)$ on the curve, the slope of the tangent line is equal to $-\frac{y}{x+y}$. 2. **Wr

1. **Problem statement:** Find the error of the 4th degree Taylor polynomial approximation for $\ln(\sec(0.4))$. 2. **Formula and explanation:** The Taylor series expansion of a fu

1. **Problem statement:** Find the first few coefficients $C_n$ of the Taylor series for $f(x) = \ln(\sec(x))$ centered at $a=0$, where the series is given by $\sum C_n x^n$. 2. **

1. **Problem Statement:** Given the function $$f(x) = \frac{x^2 - 2}{x^4}$$, find its first derivative $$f'(x)$$ and second derivative $$f''(x)$$. 2. **Rewrite the function:** To d

1. **Problem Statement:** Find the critical points, inflection points, intervals of increase/decrease, and concavity for the functions:

1. **Problem Statement:** Evaluate the integrals \(\int_0^{18} f(x) \, dx\), \(\int_0^{45} f(x) \, dx\), \(\int_{45}^{63} f(x) \, dx\), and \(\int_0^{81} f(x) \, dx\) by interpreti

1. **Problem:** Find the limit $$\lim_{x \to 1} \frac{x^3 - 3x}{x - 1}$$ using the values $$x = \{0.9, 0.99, 0.999, 1.001, 1.01, 1.1\}$$. 2. **Formula and rules:** The limit of a f

1. **Problem Statement:** Evaluate the definite integrals of the function $f(x)$ given its graph: a) $\int_{-2}^{3} f(x) \, dx$

1. **State the problem:** We need to solve for $h$ given the expression for the derivative: $$\frac{1}{3\pi} \times \frac{1}{2} h^2 \times h.$$ 2. **Simplify the expression:** Mult

1. **State the problem:** We need to sketch the curves $y = x^2 + 3$ and $y = 7 - 3x$ and find the area enclosed between them using integration. 2. **Find the points of intersectio

1. **State the problem:** We need to sketch the curves $y = x^2 + 3$ and $y = 7 - 3x$ and find the area enclosed between them using integration. 2. **Find the points of intersectio

1. We are asked to find the first and second derivatives of the function $$f(x) = \frac{1}{2 + \sin x}$$. 2. The first derivative of a function of the form $$\frac{1}{g(x)}$$ is gi

1. We are asked to find the derivative of the function $$f(x) = \sin^2 x + \frac{4}{\sin^2 x}$$ with respect to $$x$$. 2. Recall the derivative rules:

1. **Problem statement:** Show that $$\lim_{n \to \infty} \frac{1}{2n - 1} = 0$$ using the definition of the limit. 2. **Definition of limit for sequences:** For a sequence $(a_n)$

1. **State the problem:** Given the implicit equation $$x^{2} + y^{2} = A e^{B x}$$ where $A$ and $B$ are constants, find the resulting differential equation involving $y', y''$. 2

1. **State the problem:** We are given the implicit equation $$x^2 + y^2 = A e^{B x}$$ and asked to find the resulting differential equation by differentiating with respect to $x$.

1. **State the problem:** We are given two functions $y_1 = \sin(x^2)$ and $y_2 = x \cos(x^2)$ and asked to evaluate their Wronskian $W(y_1,y_2)$ at $x = \sqrt{\pi}$. 2. **Recall t