Antiderivative and Indefinite Integration

[domyy]

Homework Statement

∫(x³ - 3x² + x + 1)/√x dx

The Attempt at a Solution

∫x^3-1/2 - 3∫x^2-1/2 + ∫x^1-1/2 + ∫x^1-1/2

∫x^5/2 - 3∫x^3/2 + ∫x^1/2 + ∫^1/2

(x^7/2)7/2 - (3x^5/2)5/2 + (x^3/2)3/2 + (x^3/2)3/2 + C

(2x^7/2)/7 - (6x^5/2)/5 + 2x^3/2)/3 + (2x^3/2)/3 + C

Thank you so much!

 

[MostlyHarmless]

First, the last integral you wrote, should be $$∫{\frac{1}{√x}}$$
Second, When integrating, you divide by the new power, you are multiplying, so you should 2/7, 2/5.. etc..

Edit: I am not sure what you did on the line just after integrating, but the line after it is mostly right, except for that last part.

 

[domyy]

I don't know if I understand it.

If I have:

∫x³/√x dx

= ∫x^{3 - 1/2}
= ∫x^5/2
= (x^{5/2 + 1})/ 5/2 +1
= (x^7/2)/7/2 + C

Simplyfying:

2x^7/2/7 + C

Is this correct?

If not, what am I doing wrong?

Also, I am not sure how to proceed with ∫1/√x dx

If I have ∫1 dx:

= 1 + C

So for ∫1/√x dx I'll have:

=x/√x
=x . x^{- 1/2}
=x^1/2
=(x^{1/2+ 1})/1/2 +1 + C

 

[MostlyHarmless]

The first part was right. for ∫1/√x dx. rewrite it as ∫x^-1/2dx

EDIT: Nowhere in the equation would you have ∫1dx, but if you did. it would be x+c, not 1+c.