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More general formula for integrals 
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#1
Feb913, 12:28 PM

P: 8

I was wondering: Is there an even more general formula for the integral than int(x^k) = (x^(k+1))/(k+1) that accounts for special cases like int(x^(1)) = lnx and possibly u substitutions?



#2
Feb913, 02:44 PM

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You can combine both in a single formula:
"int(x^k) = (x^(k+1))/(k+1) for k!=1, int(x^(1))=ln(x)" Apart from that... no. 


#3
Feb913, 11:55 PM

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P: 2,264

use limits
$$\int \! x^k \, \mathrm{d}x=\lim_{a \rightarrow k+1} \frac{x^a}{a}+\mathrm{Constant}$$ That is a removable singularity. When we write it in terms of usual functions we appear to be dividing by zero, but we could define a new function without doing so. Other examples include sin(x)/x log(1+x)/x (e^x1)/x (sin(tan(x))tan(sin(x)))/x^7 going the other way we can define the function of two variables $$\mathrm{f}(x,k)=\int \! x^k \, \mathrm{d}x$$ without any worry about dividing by zero 


#4
Feb1013, 07:23 AM

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More general formula for integrals



#5
Feb1013, 10:55 AM

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#6
Feb1013, 12:22 PM

P: 759

This is a funny question !
May be, more intuitive if presented on the exponential forme, such as : 


#7
Feb1013, 02:19 PM

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