Integration and Kinematics

Integration as differentiation run backwards, definite integrals as areas, and kinematics — the one topic where the derivative and the integral both mean something you can watch happen.

หลักสูตรแกนกลาง / สสวท.: แคลคูลัส (Calculus)ตรงกับ ม.6 ปริพันธ์และการประยุกต์ (เพิ่มเติม) — ส่วนจลนศาสตร์อยู่ในวิชาฟิสิกส์ของหลักสูตรไทย ไม่ใช่คณิตศาสตร์
Common Core (US): —
Cambridge: IGCSE 0606 — Topic 14: Differentiation and integration; kinematics applications

12.1 Integration as the reverse of differentiation

The rules, each one a derivative read backwards

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + c (n \neq = - 1) \int (a x + b)^{n} d x = \frac{( a x + b ) ^{n + 1}}{a ( n + 1 )} + c

The $\frac{1}{a}$ in the second is the chain rule being undone. Forgetting it is the most frequent error in the whole chapter.

The others this syllabus expects

\int sin x d x = - cos x + c \int e^{x} d x = e^{x} + c \int \frac{1}{x} d x = ln ∣ x ∣ + c

The minus sign on the first is the one that goes missing. Differentiating your answer takes five seconds and catches it.

ข้อควรระวัง

The $+ c$ is not decoration. In a question that gives you a point on the curve, $c$ is the only thing that distinguishes the curve you want from every parallel curve above and below it — and finding it is usually the second half of the marks.

ตัวอย่าง

A curve has $\frac{d y}{d x} = 6 x^{2} - 4$ and passes through $(2, 5)$ . Find its equation.

วิธีทำ

1.Integrate: $y = 2 x^{3} - 4 x + c$
2.Substitute the point: $5 = 16 - 8 + c$
3. $c = - 3$

ตอบ $y = 2 x^{3} - 4 x - 3$

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