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Evaluate the integral using a trig substitution: `int x^3/(sqrt(4+x^2)) dx`

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Inna Shpiro eNotes educator | Certified Educator

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In this case, this would be x = 2tan(t).

This is because the the trigonometric identity `tan^2(t) + 1 = sec^2(t)` can then be applied:

`sqrt(4 + x^2) = sqrt(4 + 4tan^2(t)) = sqrt(4(tan^2(t) + 1)) `

`= sqrt(4sec^2(t)) = 2sec(t)`

Also, if x = 2tan(t), then `dx = 2sec^2(t)dt` and `x^3 = 8tan^3(t)dt`

Plugging all this into original integral, we get

`int (8tan^3(t))/(2sec(t)) 2sec^2(t)dt`

This simplifies to


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