Start by performing #color(blue)(" Integration by parts")# using the formula
#color(blue)(int u dv=uv-int v du )#
the solution is quite long because it consists of two terms.
from the given:
#int (x sin x - sec 2x) dx#
#color (red)("First part:")#
#int (x sinx) dx#
Let #u=x#, and #dv=sin x dx#
and #v=-cos x# and #du=dx#
so that #int (x sinx) dx=-x*cos x-int -cos x dx#
then #color (blue)(int (x sinx) dx=-x*cos x+sin x)#
#color (red)(" second part:")#
Use the formula: #int (sec u) du=ln (sec u+tan u)+C#
so that
#color(blue)(int sec 2x dx=1/2*ln(sec 2x+tan 2x))#
Now , combine the first and second parts
#color (magenta)(f(x)=int (x sin x - sec 2x) dx=#
#color(magenta)(-x*cos x+sin x-1/2*ln(sec 2x+tan 2x)+C)#
Solve for C: using #f(pi/12)=-2# and then
#-2=-(pi/12)*cos (pi/12)+sin (pi/12)-1/2*lnabs(sec 2*(pi/12)+tan 2*(pi/12))+C#
#-2=-(pi/12)*1/2*sqrt(2+sqrt3)+1/2*sqrt(2-sqrt3)-1/2*lnabs(2/sqrt3+1/sqrt3)+C#
#C=1/2 ln sqrt3+pi/24*sqrt(2+sqrt3)-1/2*sqrt(2-sqrt3)-2#
The final answer is
#color (magenta)(f(x)=-x*cos x+sin x-1/2 ln (sec 2x+tan 2x) + 1/2 ln sqrt3+pi/24*sqrt(2+sqrt3)-1/2*sqrt(2-sqrt3)-2)#
simplifying the square roots we have the following
#color (magenta)(f(x)=-x*cos x+sin x-1/2 ln (sec 2x+tan 2x) -1.73129)#
