∫ calculus

1. Solve the inequalities and equations: I. Solve \(|2x + 3| \leq |2x + 1|\)

1. Solve the inequalities and equations: I. Solve $$|2x + 3| \leq |2x + 1|$$

1. **State the problem:** We need to find the values of various limits and function values for the piecewise function $g(t)$ at specific points $t=0, 2,$ and $4$ from the graph pro

1. Problem: Evaluate $$\int_0^4 x^3 \sqrt{4x - x^2} \, dx$$ Step 1: Rewrite the integrand: $$\sqrt{4x - x^2} = \sqrt{-(x^2 -4x)} = \sqrt{4^2 - (x-2)^2}$$ but it's simpler to use su

1. **State the problem:** We want to evaluate the definite integral $$\int_0^4 x^3(4x - x^2)^{1/2} \, dx.$$\n\n2. **Simplify the integrand:** Notice that $$4x - x^2 = x(4 - x),$$ s

1. We need to integrate the function $$\int \csc^5 x \, dx$$. 2. Express $$\csc^5 x$$ as $$\csc^3 x \cdot \csc^2 x$$ to use identities.

1. The problem is to find the integral of the function $$dy = \frac{1}{\frac{1}{2^2} - x}$$ with respect to $x$. 2. Simplify the denominator: $$\frac{1}{2^2} = \frac{1}{4}$$, so th

1. The problem asks us to estimate $\sin\left(\frac{1}{2}\right)$ using linear approximation. 2. Linear approximation uses the formula $f(x) \approx f(a) + f'(a)(x - a)$ near a poi

1. We need to find the integral of $\cos^6 x\,dx$.\n2. Use the power-reduction formula for cosine powers: $\cos^6 x = \left(\cos^2 x\right)^3$.\n3. Recall that $\cos^2 x = \frac{1

1. **Problem f**: Find $\frac{d}{dx}(\sqrt{x} \sin x)$. Using product rule: $\frac{d}{dx} (u v) = u' v + u v'$ where $u = \sqrt{x} = x^{1/2}$ and $v = \sin x$.

1. We start with the given function: $$y = 7\cos(x^6) - \frac{1}{8}$$

1. We are given the function composition $F(x) = f(g(x))$. The problem asks for the derivative $F'(-3)$. 2. By the chain rule, the derivative of $F$ at $x$ is

1. **State the problem:** Find the equation of the tangent line to the curve $y = \sin(\sin(x))$ at the point $\left(4\pi, 0\right)$.\n\n2. **Find the derivative of $y$:$$\frac{dy}

1. Stating the problem: Evaluate the limit $$\lim_{x\to -1} \arctan\left(\frac{2x}{1-x^2}\right)$$. 2. Understand the expression inside the arctan function: $$\frac{2x}{1-x^2}$$.

1. **State the problem:** Given functions $f(x(t)) = t^2 - 3t + 2$ and $y(t) = t^3 - 4t^2 + 1$, find the derivatives $\dot{x}$ and $\dot{y}$ at $t=2$, and determine the nature (loc

1. **State the problem:** Determine whether the integral $$\int_0^1 \frac{x}{x-1} \, dx$$ converges or diverges. 2. **Analyze the integrand:** The integrand is $$f(x) = \frac{x}{x-

1. **State the problem:** Derive the equation $x^2 + 3x + 4 = 2y^2$ implicitly to find $\frac{dy}{dx}$ and then find the equation of the normal line at the point $(1, 2)$. Also, co

1. **State the problem**: We want to show using the first principle of differentiation that $$\frac{d}{dx}(\cos x) = -\sin x.$$\n\n2. **Recall the definition of derivative from fir

1. Find $\int x^2 \, dx$. 2. Evaluate $\int e^x \, dx$.

1. **State the problem:** We want to find the derivative of the function $f(x) = \frac{1}{x-2}$ using the first principle (definition of derivative). 2. **Recall the definition of

1. Integration is a fundamental concept in calculus that involves finding the integral of a function. 2. The integral represents the area under the curve of a function over an inte