How do I find the antiderivative of (x^2+1)/√x for integration?

[Checkfate]

Hi, I am trying to integrate $$\int_{1}^{2} \frac{x^{2}+1}{\sqrt{x}}$$ using the Evaluation Theorem.

So my first step is to find the antiderivative of $$\frac{x^{2}+1}{\sqrt{x}}$$.. And that is where my troubles lie.

I start by rewriting it as $$(x^{2}+1)*(x^{-1/2}}$$ but then realize that I don't know how to find the antiderivative..

I tried using the rule $$x^{n}=\frac{x^{n+1}}{n+1}$$

and got $$(\frac{x^{3}}{3}+x)*2*\sqrt{x}$$ but this does not differentiate into the original function, can someone help me out?

 

[Hootenanny]

Consider your function;

$$(x^2 +1)\cdot x^{-\frac{1}{2}}$$

Now open the parentheses.

 

[Checkfate]

Aha, got it :) Okay so generally you always want to multiply out to get addition and subtraction, right?

And I got $$\frac{2x^{5/2}}{5}+2x^{1/2}$$ which is correct :).

 

[Hootenanny]

Aha, got it :) Okay so generally you always want to multiply out to get addition and subtraction, right?

And I got $$\frac{2x^{5/2}}{5}+2x^{1/2}$$ which is correct :).

Yes, it is usually easier to multiply out the parentheses since you can integrate [or differentiate] each term individually. You could of course use integration by parts to find the integral directly from the factorised form but this would be far more complicated.

 

[Checkfate]

Okay great, thanks.

 

[Hootenanny]

Okay great, thanks.

My pleasure