Evaluate the indefinite integral using u-substitution. Use C for the constant of integration.

Question

∫x³√4+x² dx

I’ve done other problems with u-substitution that results du=dx, but I am a little lost on how to start this so any explanation to the result of the answer would be appreciative!

Answer 1

Assuming that this is your integral: ∫x³√(4+x²) dx

Let's take u = x²

This means that du = 2x dx and the dx = du / (2x)

Substituting and simplifying:

∫u*x√(4+u)*(1/(2x)) du

(1/2) * ∫u√(4+u) du

Now let's say that s = 4 + u. Then u = s - 4 and ds = du.

(1/2) * ∫(s-4)√(s) ds

Let's distribute s-4.

(1/2) * ∫(s√(s)-4√(s)) ds

(1/2) * ∫(s^3/2-4√(s)) ds

So our integral is:

(1/2) * ((2/5)s^5/2 - 4*(2/3)*s^3/2) + C

(1/2) * ((2/5)s^5/2 - (8/3)*s^3/2) + C

(1/5)s^5/2 - (4/3)*s^3/2 + C

s = 4 + u and u = x² so s = 4 + x²

So our integral is:

(1/5)(4 + x²)^5/2 - (4/3)*(4 + x²)^3/2 + C