Quod Erat Demonstrandum


Different formats of primitive functions

Filed under: HKALE,Pure Mathematics — johnmayhk @ 4:52 下午

Just take a rest from work, type something boring here…

\int \frac{dx}{\sqrt{1 - x^2}} = \sin^{-1}x + C

Students, you may regard the above as a formula or derive it by using trigonometric substitution x = \sin\theta every time.

As you may know that the expression of a primitive is not unique, we may have other forms being a primitive of \frac{1}{\sqrt{1 - x^2}}, say

\int \frac{dx}{\sqrt{1 - x^2}} = -\cos^{-1}x + C
\int \frac{dx}{\sqrt{1 - x^2}} = -2\cos^{-1}\sqrt{\frac{x+1}{2}} + C
\int \frac{dx}{\sqrt{1 - x^2}} = \sin^{-1}(a\sqrt{1 - x^2} + x\sqrt{1 - a^2}) + C where |a| \le 1

To show the validity of the above, simply by differentiation (try, esp. the last one). But, apart from the above, any other ‘format’ of a primitive? How to obtain primitives of different ‘formats’ (esp. the last one)?

1 則迴響 »

  1. 諗唔到最後果個sub咩@@…有冇提示

    迴響 由 whsvin — 2009/10/05 @ 11:15 下午 | 回覆

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