∫ calculus

1. **State the problem:** Show, using the properties of limits, that if $$\lim_{x \to 5} f(x) = 3,$$ then $$\lim_{x \to 5} \frac{x^2 - 4}{f(x)} = 7.$$

1. **State the problem:** Find the value of $k$ such that the function $$f(x) = \begin{cases} \frac{x^2 - 1}{x - 1}, & x \neq 1 \\ k, & x = 1 \end{cases}$$

1. **State the problem:** Find the derivative of the function $f(x) = x(x-7)^7$. 2. **Formula and rules:** We will use the product rule for derivatives, which states:

1. **State the problem:** We need to estimate the instantaneous rate of change of the function at $x=3$ based on the given graph points. 2. **Recall the concept:** The instantaneou

1. The problem asks to identify points where the graph is not differentiable. 2. A graph is not differentiable at points where it has sharp corners, cusps, discontinuities, or vert

1. **Énoncé du problème :** Calculer l'intégrale $$I = \int_1^e \frac{\ln(x)}{x^2} \, dx$$ par intégration par parties. 2. **Formule d'intégration par parties :** $$\int u \, dv =

1. The problem is to find the primitive (antiderivative) of the function $f(x) = \arctan(x) - \sin(x)$ on the interval $\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$. 2. The primitive

1. **State the problem:** We have a rectangular sheet of metal with width $w=200$ mm and length $l=300$ mm.

1. **Stel het probleem:** Los de onbepaalde integraal op $$\int \frac{dx}{3x^4}$$. 2. **Formule en regels:** We gebruiken de regel voor machtsfuncties: $$\int x^n dx = \frac{x^{n+1

1. **Problem statement:** We need to find the average mass of the compound remaining over the interval $0 \leq t \leq 25$ using integration. 2. **Recall the formula for average val

1. **State the problem:** We want to evaluate the integral $$\int (2x^2 + 1)^{\frac{1}{3}} x^3 \, dx.$$\n\n2. **Use u-substitution:** Let $$u = 2x^2 + 1.$$ Then, differentiate both

1. **State the problem:** We want to solve the integral $$\int (4x^2 - 12x + 9)^{\frac{2}{3}} \, dx$$ using substitution. 2. **Identify substitution:** Notice the expression inside

1. The problem is to solve the integral without using substitution. 2. The integral is not explicitly given, so let's assume a common integral that often requires substitution, for

1. **State the problem:** We want to evaluate the integral $$\int \frac{e^{2x}}{(e^{2x} - 1)^3} \, dx$$ where $x \neq 0$.

1. **State the problem:** We need to estimate the speed of the horse at 4 seconds using the gradient of the tangent to the distance-time curve at that point. 2. **Understanding the

1. **State the problem:** We need to evaluate the integral $$\int x^2 \sec^2(x^3) \, dx.$$\n\n2. **Identify the substitution:** Notice the inner function inside the secant squared

1. **State the problem:** Evaluate the integral $$\int x^2 \sec^2(x^3) \, dx$$. 2. **Identify the substitution:** Notice the inner function inside the secant squared is $x^3$. Let

1. **State the problem:** Differentiate the function $y=7x^3 - 5x^2$ with respect to $x$. 2. **Recall the differentiation rules:**

1. We are asked to find the limit $$\lim_{h \to 0} \frac{f(8+h) - f(8)}{h}$$ where $$f(x) = x^2 + 3$$. 2. This limit represents the definition of the derivative of $$f(x)$$ at $$x=

1. **State the problem:** We need to find the integral $$\int (x+20) \arctan(21x) \, dx$$. 2. **Formula and approach:** Use integration by parts, where $$\int u \, dv = uv - \int v

1. Сформулюємо задачу: знайти \( \int (x + 20) \arctan(2x) \, dx \). 2. Використаємо метод інтегрування за частинами. Формула інтегрування за частинами: