Reduction Formulas
1) \({{I}_{m,n}}=\frac{{{\sin }^{m+1}}x\ {{\cos }^{n-1}}x}{m+n}+\frac{n-1}{m+1}{{I}_{m,n-2}}\).
Proof: \({{I}_{m,n}}=\int{{{\sin }^{m}}x\ {{\cos }^{n}}x\ dx\ =\int{\left( {{\sin }^{m}}x\ {{\cos }^{n-1}}x \right)\cos x\ dx}}\),
\(={{\sin }^{m}}x\ {{\cos }^{n-1}}x\ \sin x-\int{\left[ {{\sin }^{m}}x\left( n-1 \right){{\cos }^{n-2}}x\left( -\sin x \right)+{{\cos }^{n-1}}x\ m\ {{\sin }^{m-1}}x\ \cos x \right]}\sin x\ dx\),
\(={{\sin }^{m+1}}x\ {{\cos }^{n-1}}x+\left( n-1 \right)\int{{{\sin }^{m}}x{{\cos }^{n-2}}x\ {{\sin }^{2}}x\ dx-m\int{{{\sin }^{m}}x\ {{\cos }^{n}}x\ dx}}\),
\(={{\sin }^{m+1}}x\ {{\cos }^{n-1}}x+\left( n-1 \right)\int{{{\sin }^{m}}x{{\cos }^{n-2}}x\ \left( 1-{{\cos }^{2}}x \right)dx-m\ {{I}_{m,n}}}\),
\(={{\sin }^{m+1}}x\ {{\cos }^{n-1}}x+\left( n-1 \right)\int{{{\sin }^{m}}x{{\cos }^{n-2}}x\ dx-\left( n-1 \right)\int{{{\sin }^{m}}x\ {{\cos }^{n}}x\ dx-}m\ {{I}_{m,n}}}\),
\(={{\sin }^{m+1}}x\ {{\cos }^{n-1}}x+\left( n-1 \right){{I}_{m,n-2}}-(n-1){{I}_{m,n}}-m{{I}_{m,n}}\),
\(\left( m+n \right){{I}_{m,n}}={{\sin }^{m+1}}x\ {{\cos }^{n-1}}x+\left( n-1 \right){{I}_{m,n-2}}\),
\({{I}_{m,n}}=\frac{{{\sin }^{m+1}}x{{\cos }^{n-1}}x}{m+n}+\frac{n-1}{m+n}{{I}_{m,n-2}}\).
2) \({{I}_{m,n}}=-\frac{{{\sin }^{m-1}}x{{\cos }^{n+1}}x}{m+n}+\frac{m-1}{m+n}{{I}_{m-2,n}}\).
Proof: \({{I}_{m,n}}=\int{{{\sin }^{m}}x{{\cos }^{n}}x\ dx=\int{{{\sin }^{m-1}}x{{\cos }^{n}}x\sin x\ dx}}\),
\(={{\sin }^{m-1}}x{{\cos }^{n}}x\left( -\cos x \right)-\int{\left[ {{\sin }^{m-1}}x\ n{{\cos }^{n-1}}x\left( -\sin x \right)-{{\cos }^{n}}x\left( m-1 \right){{\sin }^{m-2}}x\cos x \right]\left( -\cos x \right)dx}\),
\(=-{{\sin }^{m-1}}x{{\cos }^{n+1}}x-n\int{{{\sin }^{m}}x{{\cos }^{n}}x\ dx+\left( m-1 \right)\int{{{\sin }^{m-2}}x{{\cos }^{n}}x{{\cos }^{2}}x\ dx}}\),
\(=-{{\sin }^{m-1}}x{{\cos }^{n+1}}x-n\ {{I}_{m,n}}+(m-1)\int{{{\sin }^{m-2}}x{{\cos }^{n}}x\ \left( 1-{{\sin }^{2}}x \right)dx}\),
\(=-{{\sin }^{m-1}}x{{\cos }^{n+1}}x-n\ {{I}_{m,n}}+(m-1)\int{{{\sin }^{m-2}}x{{\cos }^{n}}x\ dx-\left( m-1 \right)\int{{{\sin }^{m}}x{{\cos }^{n}}x\ }dx}\),
\(=-{{\sin }^{m-1}}x{{\cos }^{n+1}}x-n\ {{I}_{m,n}}+(m-1){{I}_{m-2,n}}-\left( m-1 \right){{I}_{m,n}}\),
\(\left( m+n \right){{\operatorname{I}}_{m,n}}=-{{\sin }^{m-1}}x{{\cos }^{n+1}}x+(m-1){{I}_{m-2,n}}\),
\({{I}_{m,n}}=\frac{-{{\sin }^{m-1}}x{{\cos }^{n+1}}x}{m+n}+\frac{m-1}{m+n}{{I}_{m-2,n}}\).