∫ calculus

1. Solve the inequality $$\sqrt{x^2 - 3} \leq \frac{2}{\sqrt{x} - 2}.$$ Step 1: Determine the domain. Inside the square root, $x^2 - 3 \geq 0 \Rightarrow |x| \geq \sqrt{3}$. Also,

1. First, restate the problem: Differentiate the equation $$e^{x+y} - 3xy - 2 = y$$ with respect to $$x$$. 2. We will use implicit differentiation because $$y$$ is a function of $$

1. The problem is to differentiate the given equation with respect to $x$. 2. Since the equation was not explicitly provided, let's assume you want to differentiate a general funct

1. **State the problem:** Differentiate the given implicit function $$e^{x+y} - 3x + 2 = y$$ with respect to $x$. Then evaluate the derivative at the point $\left( \ln 2, 0 \right)

1. The problem is to differentiate common trigonometric functions such as $\sin x$, $\cos x$, $\tan x$, etc. 2. Recall the basic derivatives:

1. **State the problem:** We want to evaluate the integral $$\int \frac{\cosh x}{\cosh^2 x + \sinh^2 x} \, dx.$$\n\n2. **Simplify the denominator:** Recall the identity $$\cosh^2 x

1. The term "concave up" describes the shape of a graph of a function. 2. A graph is concave up if the curve bends upward like a U shape.

1. **Find the fixed points for the function $f(x) = x^2 - 6$ in the interval $[-1,4]$**. A fixed point is where $f(x) = x$. So, solve:

1. Find the fixed points for the function $f(x) = x^2 - 6$ in the interval $[-1,4]$. A fixed point is where $f(x) = x$. So solve:

1. Let us solve the integral $\int \tan^5 x \, dx$. 2. Rewrite $\tan^5 x$ as $\tan^4 x \cdot \tan x = (\tan^2 x)^2 \cdot \tan x$.

1. **Problem 12:** Find the equation of the tangent line to the curve $y = 2x \sin x$ at the point $\left(\frac{\pi}{2}, \pi\right)$. Step 1: Differentiate $y$ using the product ru

1. Stating the problem: We are given acceleration $a = \frac{dv}{dt} = 6 - 2t$, velocity $v$ is a function of $t$, and displacement $S$ defined as the integral of velocity over tim

1. **Problem statement:** Define limit and find the following limits: i. $$\lim_{x\to 1} \frac{x^3 - 3x^2 + 3x - 1}{x^3 - x}$$

1. We are asked to find the integral of $\cos^6 x \, dx$. 2. Use the power-reduction formula for cosine:

1. We are asked to integrate the function $$\int \cos(6x+4)\, dx.$$\n\n2. Recall that the integral of $$\cos(u)$$ with respect to $$u$$ is $$\sin(u) + C$$, where $$C$$ is the const

1. Stating the problem: We are given the function $y = \cos(2x)$ and asked to find its derivative $\frac{dy}{dx}$. 2. Recall the chain rule for differentiation: If $y = \cos(u)$ wh

1. The problem asks us to find the derivative of $y = \cos^2 x$.\n2. Rewrite $y = \cos^2 x$ as $y = (\cos x)^2$ to use the chain rule easily.\n3. Using the chain rule, the derivati

1. The problem is to integrate the function $$\sin^3 x \cos^4 x$$ with respect to $$x$$. 2. Begin by rewriting $$\sin^3 x$$ as $$\sin x \cdot \sin^2 x$$.

1. **State the problem:** Differentiate the function $$y = \frac{1+x}{1-x}$$ with respect to $$x$$ and find $$\frac{dy}{dx}$$. 2. **Apply the quotient rule:** For $$y = \frac{u}{v}

1. The problem is to find the integral of $\sin^3(x) \cos^5(x) \, dx$. 2. First, express the powers in a manageable form: rewrite $\sin^3(x)$ as $\sin(x) \sin^2(x)$.

1. The problem asks us to find the derivative of the function $$y=5(6-x^2)$$. 2. We start by recognizing that $$y$$ is a function of $$x$$, where $$y=5(6-x^2)$$ is a product of a c