∫ calculus

1. Differentiate $y = x \sin x$ with respect to $x$. Use the product rule: $\frac{d}{dx}[uv] = u'v + uv'$.

1. **Problem:** Solve the integral that leads to the expression $\ln(|\ln|x||) + c$. 2. **Formula and rules:** The integral involves nested logarithms. Recall that $\frac{d}{dx} \l

1. **State the problem:** We are given that $f$ is continuous and $$\int_0^9 f(x) \, dx = 10,$$

1. We are asked to evaluate the integral $$\int \frac{e^{\sqrt{6y + 4}}}{\sqrt{6y + 4}} \, dy.$$\n\n2. To solve this, use the substitution method. Let $$u = \sqrt{6y + 4}.$$\n\n3.

1. Problem statement: Compute the indefinite integrals in 4(a), evaluate the indefinite integrals in 4(b), solve the antiderivative and area problems in 5, and carry out the differ

1. **Stating the problem:** We are given the volume change rate of a balloon as $\frac{dV}{dt} = 1.08\pi$ cm³/s and asked to find the rate of change of the radius $\frac{dr}{dt}$ w

1. **Problem Statement:** Set up the integral to calculate the volume of the region bounded by the cylinder $z = y^2$, the planes $x=0$, $x=1$, $y=-1$, $y=1$, and the XY-plane ($z=

1. The problem is to differentiate the function, but the function itself was not provided. 2. To differentiate a function $f(x)$, we use the derivative rules such as the power rule

1. **State the problem:** Differentiate the function $$y = 2x \log_3(\sqrt{x})$$. 2. **Recall the formula and rules:**

1. **State the problem:** Find the derivative of the function $$y = x^{3x}$$ using logarithmic differentiation. 2. **Recall the formula and rules:** For functions of the form $$y =

1. **State the problem:** We are given the derivative of a function $f'(x) = e^{-x^2} (x^2 - 1)(2x - 3)$ and asked to find all values of $x$ where the tangent line to $f(x)$ is hor

1. **Problem Statement:** Find the slope of the tangent line to the graph of $f(x) = \tan(x)$ at $x = \frac{\pi}{6}$. 2. **Formula:** The slope of the tangent line to a function at

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{\ln(x)}{(x-1)^2}$$. 2. **Recall the formula:** To differentiate a quotient $$\frac{u}{v}$$, use the quo

1. **State the problem:** Find the limits of the function $$f(x) = \frac{\ln(x)}{(x-1)^2}$$ as $$x \to +\infty$$, $$x \to 0^+$$, $$x \to 1^+$$, and $$x \to 1^-$$. 2. **Recall impor

1. **State the problem:** We want to find the direction of change (increasing or decreasing) of the function $$f(x) = \sqrt{x} - 1 - \sqrt{x} \ln(x)$$ for $$x > 0$$ since the funct

1. **State the problem:** Find the derivative of the function $$f(x) = \sqrt{x} - 1 - \sqrt{x} \ln(x)$$. 2. **Recall the formulas and rules:**

1. **State the problem:** Find the limit of the function $$f(x) = \sqrt{x - 1} - \sqrt{x} \ln(x)$$ as $$x \to +\infty$$. 2. **Recall the behavior of components:**

1. The problem is to analyze the behavior of a function as $x$ approaches $+\infty$ (positive infinity). 2. When $x \to +\infty$, we look at the limit $\lim_{x \to +\infty} f(x)$ t

1. **State the problem:** We want to find the limit of the function $$f(x) = \sqrt{x} - 1 - \sqrt{x} \ln(x)$$ as $$x$$ approaches $$0^+$$ (from the right side). 2. **Recall the beh

1. **Problem:** Find the derivative $\frac{d}{dx}(\sec^2(x^3))$. 2. **Formula and rules:** Use the chain rule: if $y = [u(x)]^2$, then $\frac{dy}{dx} = 2u(x) \cdot u'(x)$. Also, $\

1. **State the problem:** Find the third derivative $\frac{d^3y}{dx^3}$ of the function $$y = e^{-x}(\cos 2x + \sin 2x).$$ 2. **Recall the product rule and chain rule:** For deriva