Calculus  Integrating
posted by Liam
Question: ∫(x^2)/sqrt(x^2+1)
u=x^2+1 , x^2= u1, du=2xdx
∫(u1)/sqrt(u) , expand ∫u/sqrt(u)  1/sqrt(u)
Integrate: 2/3(u^(3/2))  2u^(1/2) + c
My answer: [ 2/3(x^2+1)^(3/2)  2(x^2+1) + c ]
When I took the derivative of this to check my answer, it was not (x^2)/sqrt(x^2+1). Please help :/

Steve
The problem is that you did not have du in your integrals. You lacked the extra 2x dx in your integrands.
u = x^2+1
du = 2x dx
x = √(u1)
So you have
∫x/√(x^2+1) x dx
= 1/2 ∫√(u1)/u du
which isn't really much better
Try using a trig substitution:
x = tanθ
x^2+1 = tan^2θ + 1 = sec^2θ
dx = sec^2θ dθ
∫x^2/√(x^2+1) dx
= ∫(sec^2θ1)/secθ sec^2θ dθ
= ∫sec^3θsecθ dθ
= 1/2 secθ tanθ  1/2 log tan(π/4  θ/2)
= 1/2 x√(x^2+1)  1/2 log (√(1+x^2)  x)
= 1/2 (x√(x^2+1)  sinh^{1}x)
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