∫ calculus

1. Problem: Calculate the limit \(\lim_{n \to \infty} \frac{2n^2 + 4n + 3}{(n + 3)(\sqrt{16n^2 + 2n - 1} + n - 4)}\). 2. Formula and rules: When calculating limits involving polyno

1. **State the problem:** Evaluate the integral $$\int \frac{(1-x)^2}{x \sqrt{x}} \, dx$$. 2. **Rewrite the integrand:** Note that $$\sqrt{x} = x^{1/2}$$, so the denominator is $$x

1. **State the problem:** Find the limit $$\lim_{x \to -\infty} \frac{\sqrt{x^4 + 4x + 1}}{\sqrt{x^2 + 1}}.$$\n\n2. **Recall the formula and rules:** When dealing with limits at in

1. **State the problem:** We want to find the limit as $q \to 1$ of the expression $$\lim_{q \to 1} \left( \frac{q^n}{n-1} - \frac{1}{\ln n^n} \right)^?$$ where the exponent is not

1. **State the problem:** We need to solve the integral $$\int \frac{\sqrt{x} - x^3 e^x + x^2}{x^3} \, dx.$$\n\n2. **Rewrite the integrand:** Divide each term in the numerator by $

1. **State the problem:** We need to evaluate the integral $$\int (\sqrt{x+1} - \sqrt{x-1}) \, dx.$$\n\n2. **Recall the formula:** The integral of $$\sqrt{u}$$ with respect to $$x$

1. **Problem:** Evaluate the limit of the function $f(x) = x^2 + 3x + 2$ as $x$ approaches 1. 2. **Formula and rule:** For polynomial functions, the limit as $x$ approaches a value

1. The problem is to find the limit $$\lim_{x \to 2} (x^2 - 6x + 1).$$\n\n2. The formula for limits of polynomial functions is straightforward: since polynomials are continuous eve

1. **Stating the problem:** We are asked to find the derivative with respect to $x$ and $y$ of a very complex nested function involving trigonometric, logarithmic, exponential, and

1. **Problem:** Evaluate $$\lim_{t \to 0} \left( \frac{\sin t}{t} + 2 \right)$$ 2. **Formula and rules:** We use the special limit $$\lim_{t \to 0} \frac{\sin t}{t} = 1$$ which is

1. **Problem:** Compute the integral $$\int \frac{t^2 - 2t^4}{t^4} dt$$. 2. **Rewrite the integrand:** Simplify the expression inside the integral by dividing each term by $$t^4$$:

1. **Evaluate the limit**: $$\lim_{x \to +\infty} \frac{3 - \sqrt{x}}{3 + \sqrt{x}}$$ - When $x$ tends to infinity, $\sqrt{x}$ also tends to infinity.

1. **State the problem:** Find the equation of the tangent line to the curve $f(x) = \frac{7}{x - 2}$ at $x = 3$. 2. **Recall the formula:** The equation of the tangent line at $x

1. **Evaluate the limit**: $$\lim_{x \to +\infty} \frac{3 - \sqrt{x}}{3 + \sqrt{x}}$$ 2. **Find partial derivatives** of $$z = 2x^3 + 7x^2y - 3y + 10$$:

1. **Problem:** Find the area of the region bounded by the graph of $f(x) = x^3 - 1$ and the x-axis on the interval $[-1, 2]$. 2. **Formula:** The area bounded by the curve and the

1. **State the problem:** Find the Laplace transform of the function $$f(t) = e^{2t} (\sinh(t) + \cosh(t))$$. 2. **Recall the definitions and formulas:**

1. The problem is to identify the values of $a$, $x$, and $n$ from the given function and its derivatives. 2. The function is given as $y(x) = e^{6x^2 + 3x - 2}$.

1. **State the problem:** We need to show that the function $f(x) = x^3 - 3x^2 + 2x$ satisfies Rolle's Theorem on the interval $[0, 2]$ and then find the point $c$ in $(0, 2)$ such

1. **State the problem:** Find the equation of the tangent line to the curve $y = \ln(x + 1)$ at the point $(0, 0)$. 2. **Recall the formula for the tangent line:** The equation of

1. **Problem Statement:** Calculate the definite integral of the polynomial function $$P(\delta) = 2.2128\delta^{2} + 4.8508\delta - 0.0217$$ over the interval from 0 to 0.120. 2.

1. **Problem Statement:** Find the $n$th order derivative $\frac{d^n y}{dx^n}$ for the given functions. 2. **Formula for Exponential Function Differentiation:** For $y = e^{u(x)}$,