Can someone help me with u-sub integral?

Question

What would be best used for u-subtitution?

∫ 3x²(x³ + 1)^½ dx

Answer 1

∫ 3x² (x³ + 1)^(1/2) dx =

note that the integrand includes both the function (x³ + 1) (even if powered)

and its derivative 3x²;

thus let:

(x³ + 1) = u

differentiate both sides:

d(x³ + 1) = du →

3x²dx = du

thus, substituting, you get.

∫ [(x³ + 1)^(1/2)] 3x²dx = ∫ [u^(1/2)] du =

{u^[(1/2) +1]}/[(1/2) +1] + C =

[u^(3/2)]/(3/2) + C =

(2/3)[u^(3/2)] + C

thus, substituting back u = (x³ + 1) , you get:

∫ 3x² (x³+ 1)^(1/2) dx = (2/3) (x³ + 1)^(3/2) + C

I hope it helps...

Bye!

Answer 2

∫ 3x²(x³ + 1)^½ dx

Use u = (x³ + 1)  ---> The reason is because choosing it will right away make the problem less complicated.

Now that u = (x³ + 1) --->  Then du/dx = 3x²

So --->  du = 3x² dx

∫ 3x²(x³ + 1)^½ dx = ∫ √u  du

= ∫ √u  du

= ∫ u^½  du

= u^3/2  + C

= 2/3(x³ + 1)^3/2  + C

I can't get some of these terms to space correctly. If you are unclear on any of this, you can email me.