∫ calculus

1. **Problem:** Calculate the integral $$\int x^3 \sqrt{x} \, dx$$. **Step 1:** Rewrite the integrand using exponents: $$x^3 \sqrt{x} = x^3 x^{\frac{1}{2}} = x^{3 + \frac{1}{2}} =

1. **State the problem:** We want to find the limit $$\lim_{x \to \infty} \frac{\int_0^{2x} \sqrt{1+t^2} \, dt}{x^2}.$$\n\n2. **Recall the integral and limit concepts:** The integr

1. The problem is to evaluate the integral $$\int_0^{2x} \sqrt{1+t^2} \, dt$$ as a function of $x$. 2. We use the formula for the integral of $\sqrt{1+t^2}$:

1. **State the problem:** We need to solve the integral $$\int_0^{2x} \frac{\sqrt{1+t^2}}{x^2} \, dt$$ where the upper limit depends on $x$. 2. **Rewrite the integral:** Since $x$

1. **State the problem:** We need to solve the integral $$\int_0^2 x \frac{\sqrt{1+t^2}}{x^2} \, dt$$. 2. **Simplify the integrand:** Since $x$ is a constant with respect to $t$, w

1. We are given the function $h(x) = \ln(g(x))^3$ and need to find $h'(2)$ given $g(2) = 5$ and $g'(2) = -3$. 2. First, rewrite the function using logarithm properties: $h(x) = 3 \

1. **State the problem:** We need to find the derivative of the function $$f(x) = (2x - e^{8x}) \sin 2x$$. 2. **Recall the product rule:** For two functions $u(x)$ and $v(x)$, the

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{1}{\sin(\ln x^2)}$$. 2. **Recall the formula:** To differentiate a function of the form $$f(x) = \frac{

1. **Stating the problem:** We are given a piecewise function: $$f(z) = \begin{cases} z^2 \sin\frac{1}{z} & z \neq 0 \\ 0 & z = 0 \end{cases}$$

1. Statement of the problem. We are asked to find the derivative of the function $f(x)=3x+\frac{4}{(x^2+3x+5)^2}$.

1. **State the problem:** Calculate the limit $$\lim_{x \to 0} \frac{1 - \cos x}{x^2}$$. 2. **Recall the formula and important rules:** We use the Taylor series expansion of cosine

1. **State the problem:** Find the derivative of the function $y = \cos x$. 2. **Recall the formula:** The derivative of $\cos x$ with respect to $x$ is given by

1. **State the problem:** Calculate the limit $$\lim_{x \to 0} \frac{1 - \cos^2 x}{x^2}$$. 2. **Recall the identity:** Note that $$1 - \cos^2 x = \sin^2 x$$ by the Pythagorean iden

1. **Stating the problem:** We need to find the derivative of the implicit function defined by the equation $$\cos(x^2 + 2y) + xe^{y^2} = 1$$ with respect to $x$. 2. **Formula and

1. The problem is to find the limit $$\lim_{x \to 2} \left(2 + 5x - \frac{3x^2}{2} - x\right).\n\n2. First, simplify the expression inside the limit:\n$$2 + 5x - \frac{3x^2}{2} - x

1. **State the problem:** We need to find the derivative $\frac{dy}{dx}$ implicitly from the equation $$\cos(x^2 + 2y) + xe^{y^2} = 1.$$\n\n2. **Recall the rules:**\n- Use the chai

1. **Problem statement:** Differentiate the function $y = \sin^2 x$ with respect to $x$. 2. **Formula and rules:** Use the chain rule for differentiation. If $y = [f(x)]^2$, then $

1. Let's start by stating the problem: We want to understand how to find the limit of a function as the variable approaches infinity, which means we want to know what value the fun

1. Let's start by stating the problem: L'Hopital's Rule helps us find limits of indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ when direct substitution in a limi

1. **State the problems:** Find the limits:

1. **Problem 7:** Given the piecewise function $$f(x) = \begin{cases} x^3, & x \neq 1 \\ 0, & x = 1 \end{cases}$$