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

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

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

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

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

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