How to integrate sin ln(x)
Published On :2021-01-27 17:13:00
[smallint sin lnx{rm{d}}x] [begin{array}{l} By;Substitution\ suppose Rightarrow z = ln x\ {e^z} = x\ dz = 1/xdx\ smallint {{rm{e}}^{rm{z}}}sin z{rm{d}}z\ By;Parts\ Let Rightarrow u = sin z;;;;;;;;;;;;;;;;;;;;;;dv = {e^z}\ du = cosz;;;;;;;;;;;;;v = {e^z}\ smallint {e^z}sin z;;;dz = {e^z} cdot sin z - smallint {e^z}cos z{rm{d}}z\ By;Parts\ Let Rightarrow g = cos z;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;df = {e^z}\ dg = - sinz;;;;;;;;;;f = {e^z}\ smallint {e^{rm{z}}}sin {rm{zd}}z = {e^z}sin z - left( {;{e^z}cos z + smallint {e^z}sin z{rm{d}}z;} right)\ smallint {{rm{e}}^z}sin z{rm{d}}z = {{rm{e}}^z}sin z - {{rm{e}}^z}cos z\ smallint {{rm{e}}^z}sin z{rm{d}}z = frac{1}{2}left( {{e^z}sin z - {e^z}cos z} right) end{array}]