With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \frac x {\sin^{n - 2} a x}       | c =  }} {{eqn | r = x \sin^{-n + 2} a x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = x \map {\frac {\d} {\d x} } {\sin^{-n + 2} a x} + \paren {\frac {\d} {\d x} x} \paren {\sin^{-n + 2} a x}       | c = Product Rule }} {{eqn | r = a x \paren {-n + 2} \sin^{-n + 1} a x \cos a x + \sin^{-n + 2} a x       | c = Derivative of $\sin a x$, Derivative of Power, Chain Rule for Derivatives }} {{eqn | r = \frac {-a x \paren {n - 2} \cos a x} {\sin^{n - 1} a x} + \frac 1 {\sin^{n - 2} a x}       | c = simplifying }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = \frac 1 {\sin^2 a x}       | c =  }} {{eqn | r = \csc^2 a x       | c = Cosecant is $\dfrac 1 \sin$ }} {{eqn | ll= \leadsto       | l = v       | r = \frac {-\cot a x} a       | c = Primitive of $\csc^2 a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \frac {x \d x} {\sin^n a x}       | r = \paren {\frac x {\sin^{n - 2} a x} } \paren {\frac {-\cot a x} a}       | c = Integration by Parts }} {{eqn | o =        | ro= -       | r = \int \paren {\frac {-\cot a x} a} \paren {\frac {-a x \paren {n - 2} \cos a x} {\sin^{n - 1} a x} + \frac 1 {\sin^{n - 2} a x} } \rd x       | c =  }} {{eqn | r = \frac {-x \cos a x} {a \sin^{n - 1} a x} - \int \paren {\frac {x \paren {n - 2} \cos^2 a x} {\sin^n a x} - \frac {\cos a x} {a \sin^{n - 1} a x} } \rd x       | c = simplifying }} {{eqn | r = \frac {-x \cos a x} {a \sin^{n - 1} a x} - \paren {n - 2} \int \frac {x \cos^2 a x} {\sin^n a x} \rd x + \frac 1 a \int \frac {\cos a x} {\sin^{n - 1} a x} \rd x       | c = Linear Combination of Integrals }} {{eqn | r = \frac {-x \cos a x} {a \sin^{n - 1} a x}       | c =  }} {{eqn | o =        | ro= -       | r = \paren {n - 2} \int \frac {x \paren {1 - \sin^2 a x} } {\sin^n a x} \rd x       | c = Sum of Squares of Sine and Cosine }} {{eqn | o =        | ro= +       | r = \frac 1 a \paren {\frac {-1} {\paren {n - 2} a \sin^{n - 2} a x} }       | c = Primitive of $\sin^n a x \cos a x$ }} {{eqn | r = \frac {-x \cos a x} {a \sin^{n - 1} a x}       | c =  }} {{eqn | o =        | ro= -       | r = \paren {n - 2} \int \frac {x \rd x} {\sin^n a x} + \paren {n - 2} \int \frac {x \rd x} {\sin^{n - 2} a x}       | c = Linear Combination of Integrals and simplifying }} {{eqn | o =        | ro= +       | r = \frac 1 a \paren {\frac {-1} {\paren {n - 2} a \sin^{n - 2} a x} }       | c =  }} {{end-eqn}} This leads to: {{begin-eqn}} {{eqn | l = \paren {n - 1} \int \frac {x \rd x} {\sin^n a x}       | r = \frac {-x \cos a x} {a \sin^{n - 1} a x} + \paren {n - 2} \int \frac {x \rd x} {\sin^{n - 2} a x} - \frac 1 {a^2 \paren {n - 2} \sin^{n - 2} a x}       | c =  }} {{eqn | ll= \leadsto       | l = \int \frac {x \rd x} {\sin^n a x}       | r = \frac {-x \cos a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sin^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {x \rd x} {\sin^{n - 2} a x}       | c =  }} {{end-eqn}} {{qed}}