Question 355599
a)  {{{ int ( lnt/sqrt(t), dt, 4, 9) }}}
Use integration by parts.
Let {{{u = lnt}}}, and {{{dv = dt/sqrt(t)}}}.  Then
{{{du = dt/t}}}, and {{{v = 2sqrt(t)}}}
{{{ int ( lnt/sqrt(t), dt)  = 2sqrt(t)*lnt - int(2sqrt(t)*(dt/t))}}}
={{{2sqrt(t)*lnt - 2int(dt/sqrt(t))}}}
={{{2sqrt(t)*lnt - 4sqrt(t)}}}. Thus {{{ int (lnt/sqrt(t), dt, 4, 9) = 2sqrt(9)*ln(9)-4*sqrt(9) - 2sqrt(4)*ln(4) + 4*sqrt(4)  }}}
=12ln3-8ln2-4.


b) {{{ int ( cos(sqrt(6x)), dx ) }}}
Let {{{w = sqrt(6x)}}}.  Then {{{dt = sqrt(6x)dw/3}}}.
The integral becomes {{{(1/3)*int (wcosw, dw)}}}.  Then use integration by parts:
Let u = w, and dv = coswdw.  Then
{{{(1/3)*int (wcosw, dw) = (wsinw-int(sinw,dw))/3}}}
={{{(wsinw + cosw)/3+k}}}
={{{(sqrt(6x)*sin(sqrt(6x)) + cos(sqrt(6x)))/3+k}}}