How can I integrate x^2/(sqrt(4x^2+25)?

Answer 1

Whoa !!!!!!!, that's more difficult than I though :

Here you have :

integration(x^2 / sqrt(x^2 + 4)

you have to make a big substitution :

t^2 = 25x^-2 + 4

2tdt = -50x^-3dx

x^2 = (25 / t^2 - 4)

so you have to replace that on the integrate, that's a very big substitution :

I'll replace the values, step by step :

integrate([25 / t^2 - 4]*[ dx / sqrt(4x^2 + 25])

integrate([25 / t^2 - 4]*[ dx / tx])

-integrate( [-25 / t^2 - 4]*1 / tx* ( 2tdt / 25)*x^3

-integrate( -x^2 / t^2-4)dx

then, this will be :

integrate(-25*dt / (t^2-4)^2)

-25integrate(dt / (t^2 -4)^2)

that integration is like this :

integrate(dt / (t^2 - 4)^2) :

-t/8t^2 - 1/32*ln(t-2 / t+2)

t^2 = 25x^-2 + 4

t = sqrt( 25x^-2 + 4)

25sqrt(25x^-2 + 4) / 8*25x^-2 + 4 + 25/32*ln( sqrt( 25x^-2 + 4) - 2 / sqrt( 25x^-2 + 4) +2)

Hope that helped you, it's a very difficult integration, maybe is not that difficult, the terrible this is to operate it.

That's it

Answer 2

(-25/16)ln(sqrt(4x^2+25)+2x) +(1/8)x*sqrt(4x^2+25) +C