∫ calculus

1. **Stel het probleem vast:** We willen begrijpen wat de Leibniz-notatie is en hoe je de afgeleide van een functie kunt berekenen en interpreteren. 2. **Wat is Leibniz-notatie?**

1. **Problem Statement:** Evaluate the integral $$\int x e^{x^2} \, dx$$. 2. **Choosing the Technique:** This integral suggests using substitution because the exponent $x^2$ is a f

1. **Problem Statement:** Evaluate the limit $$\lim_{x \to 0} \frac{\sin x - x}{x^3}$$ using L'Hôpital's rule. 2. **Recall the formula and rules:** When a limit results in an indet

1. **Problem statement:** Given that $f$ is an odd function, and that $$\int_0^4 f(|x|) \, dx = 1.6,$$ we need to find:

1. **Problem statement:** A particle moves in a straight line with velocity given by $$v(t) = 1 + e^{-t} - e^{-\\sin(2t)}$$ for $$0 \leq t \leq 2$$. (a) Find the velocity at $$t=2$

1. The problem is to identify whether a given series is a p-series. 2. A p-series is a series of the form $$\sum_{n=1}^\infty \frac{1}{n^p}$$ where $p$ is a positive constant.

1. **State the problem:** We want to analyze the convergence of the infinite series $$\sum_{n=r}^\infty \frac{(n-r)!}{n!}$$ where $r$ is an integer. 2. **Rewrite the general term:*

1. **State the problem:** Find the interval of convergence of the series $$1 + \frac{2}{3}x + \frac{4}{9}x^2 + \cdots$$ 2. **Identify the general term:** The series can be written

1. Problem: Find the first derivatives of the given functions. 2. Formula: Use the product rule $\frac{d}{dx}[uv] = u'v + uv'$, chain rule $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$,

1. The problem is to find the integral of the function $y^{-2}$ with respect to $y$. 2. Recall the power rule for integration: $$\int y^n \, dy = \frac{y^{n+1}}{n+1} + C \quad \tex

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