Kalkulator SMP SMA: Solving a problem of integral lnx dx

Solving a problem of integral lnx dx

note : § as sign of integral

To solve the equation of y=§ lnx dx, we
use 2 theorems ( theorems(1) and theorems(2)) of::

     -Theorem(1) § u dv = uv- §v du (partial integral )

     -Theorem(2) § u dv = §u (dv/dx) dx

     -Theorem(3) § uv dx = § u d(§ v dx)

We can simply apply the technique using the theorems above!

Example I.
y = § lnx dx

step 1 . let u=lnx, v=x, following theorem(1)/partial integral, we obtain

§lnx dx = xlnx - § x d(lnx)

step 2. let u=x, v=lnx then dv/dx=1/x., following theorem(2) we obtain

= xlnx - § x (1/x) dx

step 3.
= xlnx - §1 dx

step 4.
= xlnx - x + c

Example II.
y = § x cosx dx

step 1. - following theorem(3), let u=x, v=cosx, § cosx dx=sinx., we obtain

§ x cosx dx = § x d(sinx)

step 2. - following theorem(1)/ partial integral, let u=x, v=sinx, we obtain

= xsinx- § sinx dx

step 3.
= xsinx- (-cosx) + c

step 4.
= xsinx + cosx + c

Good luck students

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