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# Simplified integration

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Simple derivation from basic integration expressions to f’(x) f(x) types of expressions without memorizing them

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### Simplified integration

1. 1. www.etontuition.com 6385 3133 Simple derivation from basic integration expressions to f’(x) f(x) types of expressions without memorizing them By Tan Jun Wei
2. 2. www.etontuition.com 6385 3133 This is a method of deriving f’(x) f(x) types of integration expressions from the basic integration expressions learned at O levels or from MF15. There is no need for any memorizing. Rather, simple derivations will do the trick.
3. 3. www.etontuition.com 6385 3133 Start with:  xn dx = xn+1 + cn+1 for n ≠ -1
4. 4. www.etontuition.com 6385 3133 Replace x with f(x):  f(x)n dx = f(x)n+1 + cn+1 for n ≠ -1
5. 5. www.etontuition.com 6385 3133 And multiply f’(x) in:  f(x)n dx = f(x)n+1 + cn+1f'(x) for n ≠ -1
6. 6. www.etontuition.com 6385 3133 We get our final equation:  f(x)n dx = f(x)n+1 + cn+1f'(x) for n ≠ -1
7. 7. www.etontuition.com 6385 3133 The same goes for all the following:  dx = ln (x) + c x 1  dx = ex + cex  dx = -cos x + csin x
8. 8. www.etontuition.com 6385 3133 Replace x with f(x):  dx = ln (f(x)) + c f(x) 1  dx = ef(x) + cef(x)  dx = -cos f(x) + csin f(x)
9. 9. www.etontuition.com 6385 3133 And multiply f’(x) in:  dx = ln (f(x)) + c f(x) f'(x)  dx = ef(x) + cf'(x) ef(x)  dx = -cos f(x) + cf'(x) sin f(x)
10. 10. www.etontuition.com 6385 3133 We get our final equations:  dx = ln (f(x)) + c f(x) f'(x)  dx = ef(x) + cf'(x) ef(x)  dx = -cos f(x) + cf'(x) sin f(x)