We can use integration by parts.
It states that:
intudv=uv-intvdu∫udv=uv−∫vdu
We let: u=arctan(cot(x))u=arctan(cot(x)) and dv=1dv=1
=>du=d/dx[arctan(cot(x))]⇒du=ddx[arctan(cot(x))]
=>du=1/(1+cot^2(x))*d/dx[cot(x)]⇒du=11+cot2(x)⋅ddx[cot(x)]
=>du=1/(1+cot^2(x))*-csc^2(x)⇒du=11+cot2(x)⋅−csc2(x)
=>du=-csc^2(x)/(1+cot^2(x))⇒du=−csc2(x)1+cot2(x)
v=int1dxv=∫1dx
v=xv=x
We now have:
arctan(cot(x))*x-int-csc^2(x)/(1+cot^2(x))*xdxarctan(cot(x))⋅x−∫−csc2(x)1+cot2(x)⋅xdx
=>arctan(cot(x))*x-int(x*csc^2(x))/(-1-cot^2(x))dx⇒arctan(cot(x))⋅x−∫x⋅csc2(x)−1−cot2(x)dx
Remember that:
1+cot^2(x)=csc^2(x)1+cot2(x)=csc2(x) Manimpulate this to get:
-1-cot^2(x)=-csc^2(x)−1−cot2(x)=−csc2(x)
=>arctan(cot(x))*x-int(x*cancel(csc^2(x)))/(-cancel(csc^2(x)))dx
=>arctan(cot(x))*x-int-xdx
=>arctan(cot(x))*x+intxdx
Remember that:
intx^n=(x^(n+1))/(n+1)
=>arctan(cot(x))*x+x^2/2 Do you C why this is incomplete?
=>xarctan(cot(x))+x^2/2+C That is the answer!