∫ calculus

1. Find the derivatives of the following functions: 1.a. Given $F(x) = 2x^2(3x^4 - 2)$, use the product rule: $\frac{d}{dx}[u v] = u' v + u v'$. Here, $u = 2x^2$, $v = 3x^4 - 2$.

1. Problem: Find the derivative of \(F(x) = 2x^2(3x^4 - 2)\). 2. Formula: Use the product rule for derivatives: \(\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)\).

1. The problem is to differentiate a function, but the specific function is not provided. 2. Differentiation is the process of finding the derivative of a function, which represent

1. **State the problem:** Evaluate the definite integral $$\int_3^{10} x \, dx$$. 2. **Formula used:** The integral of $$x$$ with respect to $$x$$ is given by $$\int x \, dx = \fra

1. The problem is to evaluate the definite integral $$\int_2^7 x \, dx$$. 2. The formula for the definite integral of a function $f(x)$ from $a$ to $b$ is:

1. **State the problem:** We want to evaluate the integral $$\int \frac{2x - 8}{x^2 - 8x + 32} \, dx.$$\n\n2. **Rewrite the denominator:** Notice that $$x^2 - 8x + 32 = (x - 4)^2 +

1. We are asked to find the integral $$\int \csc^{2} x \cot^{3} x \, dx$$. 2. Recall the identities and derivatives:

1. **State the problem:** Evaluate the integral $$\int \frac{dx}{x^2 + 12x + 45}$$. 2. **Rewrite the quadratic expression:** Complete the square for the denominator:

1. We are asked to evaluate the integral $$\int \frac{dx}{\sqrt{x^2 - 10x + 41}}.$$\n\n2. First, complete the square inside the square root to simplify the expression.\n\n$$x^2 - 1

1. We are asked to evaluate the integral $$\int \sin^2 x \cos^3 x \, dx$$. 2. To solve this, we use the substitution method and trigonometric identities. Notice that the powers of

1. We are asked to evaluate the integral $$\int \sin^2 x \cos^3 x \, dx$$. 2. To solve this integral, we use trigonometric identities and substitution. Important rules:

1. **Problem Statement:** Find the limit of the function $f(x)$ as $x$ approaches $-5$. 2. **Understanding Limits:** The limit $\lim_{x \to a} f(x)$ is the value that $f(x)$ approa

1. **State the problem:** We need to evaluate the definite integral $$\int_{2.1}^{2.3} \left(\cos x - \ln x + e^{-x}\right) \, dx.$$\n\n2. **Recall the integral rules:** The integr

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 $$\int_a^b 3x^2 \, dx = b^3 - a^3.$$