MHB How can I evaluate the integral using a trigonometric identity?

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karush
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Evaluate using a trig identity
$$\displaystyle
\int \dfrac{5}{x^2\sqrt{25-x^2}}\, dx$$
my first inclination to set
$u=5\sin{x}$
then
$du=5\cos{x}\, dx$ or $dx=\dfrac{du}{5\cos {x}}$
 
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$x=5\sin{t} \implies dx=5\cos{t} \, dt$

$$\int \dfrac{25\cos{t}}{25\sin^2{t}\sqrt{25-25\sin^2{t}}} \, dt$$

$$\dfrac{1}{5} \int \csc^2{t} \, dt = -\dfrac{1}{5}\cot{t}+C$$

$t=\arcsin\left(\dfrac{x}{5}\right) \implies -\dfrac{1}{5}\cot{t}=-\dfrac{\sqrt{25-x^2}}{5x}$
 
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Ok this what it was on my cell phone
 

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karush said:
Ok this what it was on my cell phone

?

here is a pic ... see if your phone renders it ok
 

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Yeah thanks

That was weird
..🐴
 
If I recall, smartphones can't read LaTeX, though that was before 2018. I don't know how it is now.
 

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