To determine

To evaluate:

∫1912xdx

Answer

∫1912xdx=ln⁡3

Explanation

1)  Concept:

The Fundamental Theorem of Calculus: Suppose f is continuous on [a, b], then

∫abfxdx=Fb-F(a), where F is antiderivative of f, that is F'=f.

2) Formulae:

i) anti derivative of 1x=ln x+C

ii)  aln⁡b=ln⁡ba

3) Calculations:

Consider,

∫1912xdx

By taking 12 outside the integral

∫1912xdx=12∫191xdx

By applying formula (i) and using the Fundamental Theorem of Calculus

12∫191xdx=12lnx91=12ln9-ln1

By simplifying the above equation

12[ln⁡9-ln⁡1]=12(ln⁡9)

Using formula (ii),

12(ln⁡9)=ln⁡912=ln9 =ln⁡3

Conclusion:

Therefore,

∫1912xdx=ln⁡3