∫ calculus

1. **Problem:** Verify that the function $f(x) = \sin(9\pi x)$ satisfies the three hypotheses of Rolle's Theorem on the interval $\left[-\frac{2}{9}, \frac{2}{9}\right]$ and find a

1. **Problem:** Find the indefinite integral \(\int 6x^4 \, dx\). 2. **Formula:** The power rule for integration states:

1. **State the problem:** Evaluate the integral $$\int \frac{1}{\sqrt{4-(x+2)^2}} \, dx$$. 2. **Recall the formula:** The integral $$\int \frac{1}{\sqrt{a^2 - u^2}} \, du = \arcsin

1. **State the problem:** Evaluate the integral $$\int \frac{1}{\sqrt{4-(x+2)}} \, dx$$. 2. **Rewrite the integral:** Notice the expression inside the square root is $$4-(x+2)$$, w

1. **State the problem:** We need to evaluate the integral $$\int \frac{x^3 + 5x^2 - 4}{x^2} \, dx$$. 2. **Rewrite the integrand:** Divide each term in the numerator by $x^2$:

1. **State the problem:** We need to evaluate the integral $$\int \frac{x^2}{1-2x^3} \, dx.$$\n\n2. **Identify a substitution:** Notice the denominator is $1-2x^3$. Its derivative

1. **State the problem:** We need to evaluate the integral $$\int \frac{8x^2}{(x^3+3)^3} \, dx.$$\n\n2. **Identify a substitution:** Notice the denominator has $(x^3+3)^3$ and the

1. **State the problem:** Evaluate the integral $$\int (1 - x) \sqrt{x} \, dx.$$\n\n2. **Rewrite the integrand:** Recall that $$\sqrt{x} = x^{\frac{1}{2}}.$$ So the integral become

1. **State the problem:** Evaluate the integral $$\int \sqrt{x^3 + 2} \cdot x^2 \, dx.$$\n\n2. **Identify substitution:** Let $$u = x^3 + 2.$$ Then, differentiate both sides with r

1. **Problem Statement:** Calculate the integral $$\int \frac{x^2}{(x+1)^3} \, dx$$ which is a typical integral involving rational functions and useful in economic modeling. 2. **F

1. **State the problem:** Differentiate $y = (\sqrt{x} + 4)(\sqrt{x} - 4)$ using the Product Rule and show that $\frac{dy}{dx} = 1$. 2. **Recall the Product Rule:** For two functio

1. **State the problem:** Find the degree 3 Taylor polynomial $T_3(x)$ of the function $$f(x) = (-7x + 109)^{\frac{5}{4}}$$ at the point $a=4$. 2. **Recall the Taylor polynomial fo

1. **State the problem:** We want to find the Taylor series expansion of the function $$f(x) = x^3$$ about the point $$x=2$$ and determine the first five coefficients $$c_0, c_1, c

1. **State the problem:** Find the limit $$\lim_{x \to -\infty} \frac{1 - x^2}{x^2 + 3}$$. 2. **Recall the rule for limits at infinity:** When $x$ approaches $\pm \infty$, the high

1. Das Problem lautet: Bestimmen Sie das Integral der Funktion $p(x) = 2e^{-0,5x}$ von $x = -1$ bis $x = 3$. 2. Die Formel für das Integral einer Exponentialfunktion $e^{ax}$ ist $

1. **Problem (i):** Given $y = \frac{x^2 - x}{e^x}$, find $\frac{dy}{dx}$ and simplify. 2. **Formula:** Use the quotient rule for derivatives: $$\frac{d}{dx}\left(\frac{u}{v}\right

1. **Stating the problem:** We want to evaluate the integral $$\int \frac{-2}{2x^2 - x^3 - x} \, dx$$. 2. **Simplify the denominator:** Factor the denominator:

1. **Problem:** Calculate the definite integral $$\int_{-2}^1 \frac{4}{x^2} \, dx$$. 2. **Formula and rules:** The integral of $$\frac{1}{x^2}$$ is $$\int x^{-2} dx = -x^{-1} + C =

1. **State the problem:** We need to evaluate the integral $$\int x e^{-2x} \, dx$$. 2. **Formula and method:** We will use integration by parts, which states:

1. **State the problem:** Find the limit as $x$ approaches $+\infty$ of the expression $$\lim_{x \to +\infty} \frac{3 - x}{\sqrt[3]{8x^3 - 3x^2 + x - 4}}.$$\n\n2. **Recall the form

1. **Problem statement:** Evaluate the integral $$\int_{-1}^1 f(x) \, dx$$ where $$f(x) = e^{\sqrt[3]{x}} - 4x$$, given that $$\int_0^1 f(x) \, dx = 3e - 8$$ and $$\int_0^{-1} f(x)