∫ calculus

1. **State the problem:** We are given the displacement function of an object: $$s(t) = t - \sin 2t$$ for $$0 \leq t \leq \pi$$.

1. **Problem statement:** Find the area of the region enclosed by the graphs of $f(x) = \cos(x^2)$ and $g(x) = e^x$ for $-1.5 \leq x \leq 0.5$. 2. **Formula and approach:** The are

1. **State the problem:** We have the function $f(x) = 2x^2$ and a tangent line $T$ at $x=1$. We need to:

1. **State the problem:** Find the limit $$\lim_{x \to 2} \frac{8x^2 + 6x - 1}{\sqrt{x^2 + 5x + 2}}.$$\n\n2. **Substitute $x=2$ directly:**\nCalculate numerator: $8(2)^2 + 6(2) - 1

1. **Problem statement:** Given the function $p(x) = g(x+1) - g(x)$ and $p(x) = 2$ for all $x$, determine which statements about $g$ must be true. 2. **Understanding the problem:**

1. **Problem:** Find the limit $$\lim_{x \to a} \frac{x^3 - a^3}{a^6 - x^6}$$ where $a \neq 0$. 2. **Formula and rules:** Recall the difference of cubes factorization:

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