Can integration by sustitution only be used on definite integrals?

Question

I need to evaluate the following indefinite integral:

x(x^2+3)^(1/2)

I first thought of trying integration by parts, but I think I'm supposed to use substitution. However, all my examples have a definite integral. So what do I do?

Answer 1

I = ∫ (x) (x² + 3)^(1/2) dx
Let u = x² + 3
du = 2x dx
du / 2 = x dx
I = (1/2) ∫ u^(1/2) du
I = (1/2) u^(3/2) / (3/2) + C
I = (1/3) (x² + 3)^(3/2) + C

Answer 2

First of all, integration by substitution can be used for both definite and non-definite integrals.
x(x^2+3)^1/2
let x^2+3 = t
2xdx = dt
x dx = (1/2)dt
the given integral becomes
(1/2) integ sqrt(t) dt
(1/2)t^(1/2+1)/(1/2+1)
=(1/2)t^(3/2)/(3/2)
=(1/3)t^3/2
=(1/3)(x^2+3)^(3/2)+C

Answer 3

Let u = x^2+3
Then du/2 = xdx
So 1/2 integral u^1/2 du = (1/2)(2/3) u^3/2= 1/3(x^2+3)^3/2+C