Find the indefinite integral of the given problem

chwala
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Homework Statement
see attached
Relevant Equations
##\dfrac{d}{dx} \left[\tan^{-1}x\right]= \dfrac{1}{x^2+1}##
Now the steps to solution are clear to me...My interest is on the constant that was factored out i.e ##\frac{2}{\sqrt 3}##...

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the steps that were followed are; They multiplied each term by ##\dfrac{2}{\sqrt 3}## to realize,

##\dfrac{2}{\sqrt 3}\int \dfrac{dx}{\left[\dfrac{2}{\sqrt 3}⋅\dfrac{2x+1}{2}\right]^2+\left[\dfrac{2}{\sqrt 3}⋅\dfrac{\sqrt 3}{2}\right]^2} ##

...

Correct? Is there a different approach guys?

Now, bringing me to my question...see problem below;

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Are they missing ##\dfrac{1}{2}## somewhere?? In this case we are dividing each term by ##2^2##... we ought to have;

... ##\dfrac{1}{2}\sin^{-1} \left[\frac{x+1}{2}\right] + c##

I hope i did not overlook anything...
 
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<br /> \int \frac{1}{\sqrt{2^2 - (x + 1)^2}}\,dx = \int \frac{1}{2\sqrt{1 - u^2}}(2\,du) with u = (x + 1)/2.
 
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pasmith said:
<br /> \int \frac{1}{\sqrt{2^2 - (x + 1)^2}}\,dx = \int \frac{1}{2\sqrt{1 - u^2}}(2\,du) with u = (x + 1)/2.
Thanks ...cheers man!
 
Aaaargh! because of the square root sign! That calls for 'change of variable' ...was wondering why...great day!
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...

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