Resolución de una integral definida, como se puede solucionar

$$\begin{align}&\int_ \frac{-\pi} 4^\frac {\pi} 4( \theta- csc \theta·cotg \theta) d \theta=\\&\\&\int_ \frac{-\pi} 4^\frac {\pi} 4( \theta - \frac 1 {\sin \theta}· \frac{\cos \theta}{\sin \theta}) d \theta=\\&\\&\int_ \frac{-\pi} 4^\frac {\pi} 4( \theta-\sin^{-2} \theta \cos \theta) d \theta=\\&\\& \frac {\theta^2} 2- \frac{ \sin^{-1} \theta}{-1} \Bigg |_ \frac{-\pi} 4^\frac {\pi} 4=\\&\\&\frac {\theta ^2} 2+ \frac 1 {\sin \theta} \Bigg|_ \frac{-\pi} 4^\frac {\pi} 4=\\&\\&\frac {\theta ^2} 2+ \frac 1 {\sin \theta} \Bigg|_ \frac{-\pi} 4^0+\frac {\theta ^2} 2+ \frac 1 {\sin \theta} \Bigg|_0^\frac {\pi} 4=\\&\\&0+ \lim_{\theta \to 0^-} \frac 1 {\sin \theta}- \frac{\pi^2}{32}+( \frac 2 {\sqrt 2 } )^2+\bigg( \frac{\pi^2}{32}+( \frac 2 {\sqrt 2} )^2-0-\lim_{\theta \to 0^+} \frac 1 {\sin \theta} \Bigg)=\\&\\&- \infty- \infty= -\infty\\&\\&\end{align}$$

;)
Hola recce !
Eso es una integral impropia ya que hay un punto (theta=0) donde la integral no está definida: