Integral of irrational function

Integral of irrational function

Postby Guest » Fri Nov 25, 2016 3:48 pm

Integral of an irrational function:
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Re: Integral of irrational function

Postby Guest » Fri Nov 25, 2016 3:52 pm

a is a positive real number.
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Re: Integral of irrational function

Postby Guest » Sat Nov 26, 2016 4:39 am

Both solutions are correct, because
[tex]\arcsin(x/a) = \arctan(x/\sqrt{a^2-x^2})[/tex].

You can prove this by considering a right angled triangle with hypotenuse [tex]a[/tex], opposite side [tex]x[/tex], and by Pythagoras the adjacent side is [tex]\sqrt{a^2-x^2}[/tex]. So [tex]\sin\theta = x/a[/tex], and [tex]\tan\theta = x/\sqrt{a^2-x^2}[/tex], which means we have two ways of calculating [tex]\theta[/tex] which results in [tex]\theta = \arcsin(x/a) = \arctan(x/\sqrt{a^2-x^2})[/tex].

See also
https://en.wikipedia.org/wiki/Inverse_t ... _functions

Hope this helped,

R. Baber.
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