Reduction Formulas – II

Reduction Formulas – II

1) If \({{I}_{n}}=\int{{{\sin }^{n}}x\,dx}\) then \({{I}_{n}}=\frac{-{{\sin }^{n-1}}x\cos x}{n}+\frac{n-1}{n}{{I}_{n-2}}\), where n is positive integer.

Proof: \({{I}_{n}}=\int{si{{n}^{n-1}}x\,sin\,x\,dx}\),

\(={{\sin }^{n-1}}x\left( -\cos x \right)-\int{-\cos x\left( n-1 \right){{\sin }^{n-2}}x\,\cos x\,dx}\),

\(=-{{\sin }^{n-1}}x\,\cos x+\left( n-1 \right)\int{{{\sin }^{n-2}}x\,{{\cos }^{2}}x\,dx}\),

\(=-{{\sin }^{n-1}}x\,\cos x+\left( n-1 \right)\int{{{\sin }^{n-2}}x\,\left( 1-{{\sin }^{2}}x \right)\,dx}\),

\(=-{{\sin }^{n-1}}x\,\cos x+\left( n-1 \right)\int{{{\sin }^{n-2}}x\,dx}-\left( n-1 \right)\int{{{\sin }^{n}}x\,dx}\),

\(=-{{\sin }^{n-1}}x\,\cos x+\left( n-1 \right)\,{{I}_{n-2}}-\left( n-1 \right){{I}_{n}}\),

\({{I}_{n}}\left( 1+n-1 \right)=-{{\sin }^{n-1}}x\,\cos x+\left( n-1 \right)\,{{I}_{n-2}}\),

∴ \({{I}_{n}}=-\frac{{{\sin }^{n-1}}x\,\cos x}{n}+\frac{n-1}{n}{{I}_{n-2}}\).

2) If \({{I}_{n}}\,=\int{{{\cos }^{n}}x\,dx}\) then \({{I}_{n}}=\frac{{{\cos }^{n-1}}x\,\sin x}{n}+\frac{n-1}{n}{{I}_{n-2}}\).

Proof: \({{I}_{n}}\,=\int{{{\cos }^{n-1}}x\cos x\,dx\,=\,{{\cos }^{n-1}}x\,\sin x-\int{\sin x(n-1){{\cos }^{n-2}}x(-\sin x)dx}}\),

\(={{\cos }^{n-1}}x\,\sin x+\left( n-1 \right)\int{{{\cos }^{n-2}}x\left( 1-{{\cos }^{2}}x \right)dx}\),

\(={{\cos }^{n-1}}x\,\sin x+\left( n-1 \right)\int{{{\cos }^{n-2}}\,x\,dx-\left( n-1 \right)}\int{{{\cos }^{n}}x\,dx}\),

\(={{\cos }^{n-1}}x\,\sin x+\left( n-1 \right){{I}_{n-2}}-\left( n-1 \right){{I}_{n}}\),

\({{I}_{n}}\left( 1+n-1 \right)={{\cos }^{n-1}}x\,\sin x+\left( n-1 \right){{I}_{n-2}}\),

∴ \({{I}_{n}}=\frac{{{\cos }^{n-1}}x\,\sin x}{n}+\frac{n-1}{n}{{I}_{n-2}}\).