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

Inna Shpiro | 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

`int...

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