Calculus Integration Problem?

Question

For this problem I had to take the derivative of an equation to find the arc length. Right now I am working on the integration portion. If you could just point me in the right direction I would appreciate it.
Integrate:
sqrt(1+( ((x-3)/(6*sqrt(x))) + (sqrt(x)/3) )^2 ) dx

Thank you.

Answer 1

∫√(1+((x-3)/(6√x) + √x/3)²) dx

Start by expanding the squared term:

∫√(1+(x²-6x+9)/(36x) + (x-3)/9 + x/9) dx

Now, expand the individual terms:

∫√(1+x/36 - 1/6 + 1/(4x) + x/9 - 1/3 + x/9) dx

Combine like terms:

∫√(x/4 + 1/2 + 1/(4x)) dx

Factor out the √(1/(4x)):

∫√(x² + 2x + 1)/√(4x) dx

Of course this becomes:

1/2 ∫√(x² + 2x + 1)/√x dx

Which now factors easily as:

1/2 ∫(x+1)/√x dx

Expanding:

1/2 ∫√x + 1/√x dx

Integrating:

x^(3/2)/3 + √x + C

And we are done.

Answer 2

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