Reduction Formulas

Reduction Formulas

1) If $${{I}_{n}}=\int{{{\cot }^{n}}x\,dx}$$ then $${{I}_{n}}=\frac{-{{\cot }^{n-1}}x}{n-1}-{{I}_{n-2}}$$.

Proof: $${{I}_{n}}=\int{{{\cot }^{n-2}}x\,{{\cot }^{2}}x\,dx}=\int{{{\cot }^{n-2}}x\left( \cos e{{c}^{2}}x-1 \right)dx}$$,

$$=\int{{{\cot }^{n-2}}x\,\cos e{{c}^{2}}x\,dx-\int{{{\cot }^{n-2}}x\,dx}}$$,

$$=-\frac{{{\cot }^{n-1}}x}{n-1}-{{I}_{n-2}}$$.

2) If $${{I}_{n}}=\int{{{\sec }^{n}}x\,dx}$$ then $${{I}_{n}}=\frac{{{\sec }^{n-2}}\,x\,\tan x}{n-1}+\frac{n-2}{n-1}{{I}_{n-2}}$$.

Proof: $${{I}_{n}}=\int{{{\sec }^{n}}x\,dx\,\,\,\,=\int{{{\sec }^{n-2}}x{{\sec }^{2}}x\,dx}}$$,

$$={{\sec }^{n-2}}x\,\tan x-\int{\tan x\left( n-2 \right){{\sec }^{n-3}}x\ \sec x\ \tan x\ dx}$$,

$$={{\sec }^{n-2}}x\,\tan x-\left( n-2 \right)\int{{{\sec }^{n-2}}x\ {{\tan }^{2}}x\ dx}$$,

$$={{\sec }^{n-2}}x\,\tan x-\left( n-2 \right)\int{{{\sec }^{n-2}}x\ \left( {{\sec }^{2}}x-1 \right)\ dx}$$,

$$={{\sec }^{n-2}}x\,\tan x-\left( n-2 \right)\int{{{\sec }^{n}}x\ dx+\left( n-2 \right)\int{{{\sec }^{n-2}}x\ dx}}$$,

$$={{\sec }^{n-2}}x\,\tan x-\left( n-2 \right){{I}_{n}}+\left( n-2 \right){{I}_{n-2}}$$,

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

∴ $${{I}_{n}}=\frac{{{\sec }^{n-2}}x\ \tan x}{n-1}+\frac{n-2}{n-1}{{I}_{n-2}}$$.

3) If $${{I}_{n}}=\int{\cos e{{c}^{n}}x\ dx}$$ then $${{I}_{n}}=\frac{-\cos e{{c}^{n-2}}x\ \cot x}{n-1}+\frac{n-2}{n-1}{{I}_{n-2}}$$.

Proof: $${{I}_{n}}=\int{\cos e{{c}^{n-2}}x\cos e{{c}^{2}}x\ dx}$$,

$$=-\cos e{{c}^{n-2}}x\ \cot x-\int{-\cot x\left( n-2 \right)\cos e{{c}^{n-3}}x\left( -\cos ecx\ \cot x \right)dx}$$,

$$=-\cos e{{c}^{n-2}}x\ \cot x-\left( n-2 \right)\int{\cos e{{c}^{n-2}}x\ {{\cot }^{2}}x\ dx}$$,

$$=-\cos e{{c}^{n-2}}x\ \cot x-\left( n-2 \right)\int{\cos e{{c}^{n-2}}x\ \left( \cos e{{c}^{2}}x-1 \right)\ dx}$$,

$$=-\cos e{{c}^{n-2}}x\ \cot x-\left( n-2 \right)\int{\cos e{{c}^{n}}x\ dx+\left( n-2 \right)\int{\cos e{{c}^{n-2}}x\ dx}}$$,

$$=-\cos e{{c}^{n-2}}x\ \cot x-\left( n-2 \right)\ {{I}_{n}}+\left( n-2 \right){{I}_{n-2}}$$,

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

$${{I}_{n}}=-\frac{\cos e{{c}^{n-2}}x\ \cot x}{n-1}+\frac{n-2}{n-1}{{I}_{n-2}}$$.