Integrate 2*sqrt(x^2+1) =

Hi, guys. Could you help me to solve this:

integrate 2*sqrt(x^2+1)

Thanks!

sqrt(1 + x^2) * dx

x = tan(t)
dx = sec(t)^2 * dt

sqrt(1 + tan(t)^2) * sec(t)^2 * dt
sqrt(sec(t)^2) * sec(t)^2 * dt
sec(t) * sec(t)^2 * dt

u = sec(t)
du = sec(t) * tan(t)
dv = sec(t)^2 * dt
v = tan(t)

int(sec(t)^3 * dt) = sec(t) * tan(t) - int(tan(t) * sec(t) * tan(t) * dt)
int(sec(t)^3 * dt) = sec(t) * tan(t) - int(tan(t)^2 * sec(t) * dt)
int(sec(t)^3 * dt) = sec(t) * tan(t) - int((sec(t)^2 - 1) * sec(t) * dt)
int(sec(t)^3 * dt) = sec(t) * tan(t) - int(sec(t)^3 * dt) + int(sec(t) * dt)
2 * int(sec(t)^3 * dt) = sec(t) * tan(t) + int(sec(t) * dt)
2 * int(sec(t)^3 * dt) = sec(t) * tan(t) + ln(sec(t) + tan(t)) + C
int(sec(t)^3 * dt) = (1/2) * (sec(t) * tan(t) + ln(sec(t) + tan(t)) + C)

x = tan(t)
sec(t)^2 - tan(t)^2 = 1
sec(t)^2 - x^2 = 1
sec(t)^2 = 1 + x^2
sec(t) = sqrt(1 + x^2)

2 * (1/2) * (sqrt(1 + x^2) * x + ln(sqrt(1 + x^2) + x)) + C)
sqrt(1 + x^2) * x + ln(sqrt(1 + x^2) + x)) + C

Use the substitution x=sinh t. Then dx = cosh t dt.
Since cosh^2 - sinh^2 = 1 we have x^2 + 1 = sinh^2(t) + 1 = cosh^2(t) and
therefore
sqrt(x^2+1) = sqrt(cosh^2(t)) = |cosh(t)| = cosh(t)
since cosh is a positive function.
Therefore the integral 2*sqrt(x^2+1) dx equals the integral 2*cosh^2(t) dt = (1/2)(e^(2t)+2+e^(-2t)) dt
= (1/4)(e^(2t)-e^(-2t)) + t + C
You can express t in terms of x using the quadratic formula to get the final answer. But you can do it by yourself.