∫ calculus

1. The problem is to find the derivative of the function given as $$f'(x) = 9\sqrt{x + 5} - 9\sqrt{x - 1}$$. 2. Recall the derivative rule for a square root function: $$\frac{d}{dx

1. **State the problem:** We need to evaluate the definite integral $$\int_{-\sqrt{6}}^{\pi} (2x + 2|x|) \sin(6x) \, dx.$$\n\n2. **Understand the absolute value:** The function ins

1. **Problem statement:** Perform a curve discussion (Kurvendiskussion) for the function $$f(x) = \frac{4x}{x^2 - 4}$$. 2. **Domain:** The denominator cannot be zero, so solve $$x^

1. **State the problem:** We are given gold production data $G = f(t)$ for years $t$ from 2014 to 2018 and asked to analyze the derivative $f'(t)$, which represents the rate of cha

1. The problem asks to identify the correct graph of the derivative of the function $f(x)$, where $f(x)$ is a downward-opening parabola with vertex at approximately $(1,4)$. 2. Rec

1. The problem is to understand the behavior of a function as the input approaches negative infinity. 2. When analyzing limits at negative infinity, we look at the value of the fun

1. **State the problem:** Find the limit $$\lim_{x \to \infty} \frac{\sqrt{1 + 4x^6}}{2 - x^3}$$ as $x$ approaches infinity. 2. **Recall the rule for limits at infinity involving r

1. The problem is to find the integral of $x\sec^2(x)\,dx$. 2. We use integration by parts formula: $$\int u\,dv = uv - \int v\,du$$.

1. **State the problem:** Find the first derivative of the function $$y = \ln\left(\frac{3 - x}{(2x + 3)(x + 0)}\right)$$. 2. **Recall the formula:** The derivative of $$\ln(u)$$ w

1. **State the problem:** We need to find the integral $$\int (x+2)(5x-2)^3 \, dx$$.

1. **State the problem:** We want to evaluate the integral $$\int x^2 \cos\left(x^3\right) \, dx.$$\n\n2. **Identify the method:** This integral suggests a substitution because the

1. **State the problem:** Evaluate the definite integral $$\int_1^2 \frac{1}{x^2} \, dx$$. 2. **Recall the formula:** The integral of $$x^n$$ with respect to $$x$$ is $$\frac{x^{n+

1. **State the problem:** We want to find the derivative of the function $$f(t) = \left(1 - t\right)^2$$ raised to the power of $$-1$$, which is $$f(t) = \left((1 - t)^2\right)^{-1

1. **State the problem:** We want to find the derivative of the function $$f(x) = e^{x^2 + 3x}$$. 2. **Recall the formula:** The derivative of an exponential function with base $e$

1. **State the problem:** We are given the graph of $f'$, the derivative of $f$, and asked to determine all possible inferences about $f$ and $f'$ at $x=0$. 2. **Recall key concept

1. **State the problem:** Find the derivative $F'(x)$ of the function $f(x) = -3x^2 - x$ using the difference quotient method. 2. **Recall the difference quotient formula:**

1. **State the problem:** Find the area of the region enclosed by the graphs of $y=6x$ and $y=5x^4$. 2. **Find the points of intersection:** Set $6x=5x^4$ to find where the curves

1. **Stating the problem:** We are asked to solve an optimization problem using Lagrangian functions. This typically involves maximizing or minimizing a function $f(x,y,\ldots)$ su

1. The problem is to find the derivative $y'$ of a function $y$. 2. To find $y'$, we need the explicit form of the function $y=f(x)$, which is not provided.

1. **Problem 1:** Show that $$\int_0^1 \frac{1}{\sqrt{x^2 + 3}} \, dx = \frac{1}{2} \ln 3$$. 2. **Step 1:** Use the substitution $$u = x + \sqrt{x^2 + 3}$$.

1. The problem is to understand the basics of calculus, which involves studying rates of change and accumulation. 2. The fundamental concepts include derivatives and integrals.