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Indefinite integrals

Integration

Japanese school year: Math II

What you learn

This topic explores finding indefinite integrals with integration constants, serving as the reverse operation of differentiation. It forms the foundational calculation technique for reconstructing total quantities from rates of change and computing definite integrals. A solid grasp of polynomial differentiation rules and algebraic manipulation is required before beginning.

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Key points

This basic formula integrates the power function xnx^n (where n≠−1n \neq -1). Divide by the new exponent n+1n+1, raise xx to n+1n+1, and add an integration constant CC.

∫xn dx=1n+1xn+1+C(n≠−1)\int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \quad (n \neq -1)

This equation shows that differentiating an indefinite integral returns the original function f(x)f(x). It confirms that differentiation and integration are inverse operations.

ddx∫f(x) dx=f(x)\frac{d}{dx} \int f(x) \, dx = f(x)

This property shows that when integrating a sum f(x)+g(x)f(x) + g(x), you can integrate each function separately and then add the results together.

∫{f(x)+g(x)} dx=∫f(x) dx+∫g(x) dx\int \{f(x) + g(x)\} \, dx = \int f(x) \, dx + \int g(x) \, dx

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