Frank
Hello, this is speaking.
Today, we’ll revisit the fundamentals of indefinite integrals and strengthen your understanding through simple, approachable examples. Even if math wasn’t your strongest subject, this lesson is designed to help you move smoothly from concept to practice.
• Basic Formulas for Indefinite Integrals (with \(C\) as the constant of integration)
- Integral of a constant: \(\int kdx = kx + C\)
- Integral of \(x^{n}\): \(\int x^{n}dx = \frac{1}{n+1}x^{n+1} + C\)
- Integral of \((ax + b)^{n}\): \(\int(ax + b)^{n}dx = \frac{1}{a(n + 1)}(ax + b)^{n+1} + C\)
(Source: High School Mathematics—Once Again, p.322)
Before moving on, take a moment to review the formulas above and remember that integration is essentially the reverse operation of differentiation. In high school mathematics, this perspective appears frequently, and indefinite integrals form one of its most essential foundations.
Now, let’s try a basic practice problem.
【Problem】
Find the indefinite integral of \(3x^2 + 8\). Be sure to include the constant of integration \(C\) as needed.
【Solution】
We apply the formula \(\int x^{n}dx = \frac{1}{n+1}x^{n+1}\) and integrate each term separately.
\[
\int(3x^{2} + 8)dx
= 3 \cdot \frac{1}{3}x^{3} + 8x + C
= x^{3} + 8x + C
\]
As shown, always remember to include the constant of integration \(C\) at the end. It’s a detail that’s surprisingly easy to overlook.
As someone who originally majored outside the sciences, I still occasionally find myself confusing differentiation and integration—which I find strangely amusing. Yet, spending quiet moments writing these math-focused blog posts has become a refreshing part of my daily routine.
The older we get, the more we realize that learning has no end—and the ways we enjoy learning continue to evolve. In the next lesson, we’ll take on a slightly more advanced problem, so stay tuned!
Explore more in Mathematics for Global Business here: Continue Learning b＾＾)
【Reference Book】『High School Mathematics—Once Again』 (Author: Kazuo Takahashi / Nihon Jitsugyo Publishing)
【Lessons】Interested in online English lessons? Click here.
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