Evaluate the following integral by making the given substitution

ʃx²√(x³+1)dx, u = x³+1

Let u = x³ + 1.
So, du = 3x² dx ==> x² dx = du/3.

Hence, ʃ x² √(x³ + 1) dx
= ʃ √u du/3
= ʃ (1/3) u^(1/2) du
= (1/3) * (2/3)u^(3/2) + C
= (2/9) (x³ + 1)^(3/2) + C.

I hope this helps!