∫ calculus

1. **State the problem:** We want to determine the convergence of the infinite series $$\frac{x}{1 \cdot 2} + \frac{x^2}{3 \cdot 4} + \frac{x^3}{5 \cdot 6} + \cdots$$

1. The problem is to find the integral of $\cosh\left(\frac{x}{a}\right)$ with respect to $x$. 2. Recall that the integral of $\cosh(u)$ with respect to $u$ is $\sinh(u) + C$.

1. Problem: Evaluate the definite integral $$\int_{0}^{1} 5 e^{2x-1} dx$$. 2. Step 1: Factor constants and simplify the integrand by writing $5 e^{2x-1}=5 e^{-1} e^{2x}$.

1. **State the problem:** Evaluate the definite integral $$\int_0^1 \frac{5}{e^{2x-1}} \, dx$$. 2. **Rewrite the integrand:** Since $$\frac{5}{e^{2x-1}} = 5 e^{-(2x-1)} = 5 e^{-2x+

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{2x - 6}{\sqrt{x+1} - 2}$$. 2. **Identify the indeterminate form:** Substitute $x=3$ directly:

1. The problem involves using differentiation to find gradients, tangents, normals, stationary points (excluding points of inflexion), connected rates of change, small increments,

1. The problem is to know and use the derivatives of standard functions: $x^n$ (for any rational $n$), $\sin x$, $\cos x$, $\tan x$, $e^x$, and $\ln x$, with examples. 2. Derivativ

1. **Stating the problem:** Find the equation of the tangent line to a curve at a given point. 2. **Understanding the tangent line:** The tangent line to a curve at a point touches

1. The problem is to find the equation of the tangent line to a curve at a given point. 2. Suppose the curve is given by a function $y=f(x)$ and we want the tangent line at $x=a$.

1. The problem is to understand the concept of differentiation in calculus. 2. Differentiation is the process of finding the derivative of a function, which represents the rate of

1. The problem is to differentiate the function $$f(x) = \frac{(x^2 + 1)^2}{x^2 - 1}$$ with respect to $x$. 2. First, identify the numerator and denominator:

1. **State the problem:** We need to find the exact value of $a$ where the curve $y = \cos x \sqrt{\sin 2x}$ has a maximum point $M$ for $0 \leq x \leq \frac{\pi}{2}$. 2. **Write t

1. **Тодорхойлолт:** Дуу бичлэгийн компани ТВ сурталчилгааны өдрийн тоог $t$ гэж үзвэл, $t$ хоногийн дараах CD худалдан авах хувь нь $$1 - e^{-0.06t}$$ байна.

1. The problem is to find the integral $$\int x^3 \cos(x^4 + 2) \, dx$$. 2. Notice that the argument of the cosine function is $$x^4 + 2$$, and its derivative is $$4x^3$$, which is

1. The problem is to differentiate the function $$f(x) = x^4 + 2x^3 + x^2$$ with respect to $$x$$. 2. Recall the power rule for differentiation: $$\frac{d}{dx} x^n = nx^{n-1}$$.

1. **State the problem:** We want to evaluate the integral $$I = \int_1^4 \frac{\sqrt{x-1}}{2(x+\sqrt{x})} \, dx$$ using the substitution $u = \sqrt{x}$. 2. **Substitution:** Let $

1. The problem is to find the original function $F(x)$ given its derivative $F'(x) = 3 + \frac{5x^2 + 2}{x^{1/2}}$. 2. Rewrite the derivative to simplify the integral:

1. **State the problem:** Evaluate the definite integral $$- \int_0^1 z e^{z^2} \, dz$$. 2. **Rewrite the integral:** The negative sign can be factored out:

1. The problem is to find $\frac{dy}{dx}$ for the parametric equations $y = t^6 - 5$ and $x = 4^3 - 1$. 2. First, note that $x = 4^3 - 1 = 64 - 1 = 63$ is a constant, independent o

1. The problem is to find the derivative $\frac{dy}{dx}$ of a function $y$ with respect to $x$. 2. To proceed, please provide the explicit function $y=f(x)$ you want to differentia

1. **State the problem:** Find the derivative of the implicit function defined by the equation $$2y^3 + 3x^5 = 5y - 6$$ with respect to $x$. 2. **Differentiate both sides with resp