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I have the following integral that i want to solve:

$\displaystyle\int _{0}^{y} {(1+x)^{-.5k-u+b}dx}$

i noticed this series representation of :

$(1+x)^r=\displaystyle\sum_{m=0}^{\infty} {\frac{(-r)_m}{m!}(-x)^m}$, i found it in this website: http://mathworld.wolfram.com/BinomialSeries.html

My question is: can i use this series representation to replace the term in my integral? does it apply to my term?

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    If it is in the radius of convergence of the binomial expansion, then yes. – 2017-01-30

2 Answers 2

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You should not do that, no.

Please tell me:

  • what is the derivative of $x^{\alpha}$?
  • What is the derivative of $(1+x)^\alpha$?
  • What is the derivative of $\log x$?
  • What is the derivative of $\log (1+x)$?
  • What does the fundamental theorem of calculus say?

Now, do you see how to proceed?

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    Then, how can i solve such an integral? – 2017-01-30
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    These are hints on how to do that. Can you answer the questions? Here is another hint: If your exponent is $-1$ then you have $$\int_0^y (1+x)^{-1}dx= \log(1+y)$$ – 2017-01-30
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Simply notice that

$$\frac d{dx}(1+x)^n=n(1+x)^{n-1}$$

thus,

$$\int(1+x)^n\ dx=\frac{(1+x)^{n+1}}{n+1}+c$$

and plug in the bounds. The special case is $n=-1$, whereupon the antiderivative is the natural logarithm.

Notice, however, that your series expansion is only valid if $y$ and $0$ are within the radius of convergence, which depends on the exponents.