Reduction Formula

Reduction formula in mathematics is generally used for solving integration of higher order. Integration involving higher-order terms is difficult to handle and solve. So, to simplify the solving process of higher-order terms and get rid of the lengthy expression-solving process of higher-order degree terms – Integration processes can be simplified by using Reduction Formulas.

Reduction formulas are used to simplify the higher-order terms. Integration of higher-order terms consisting of logarithmic, algebraic, and trigonometric functions are simplified by reduction formulas. In the Reduction formula, higher-order degree terms are given a degree n. Reduction formulas with degree n are derived from the integration base formulas. All rules of integration apply to these reduction formulas as well.

Reduction Formula for different expressions are listed below:

For logarithmic functions, reduction formulas are:

• ∫ logⁿx dx = xlogⁿx -n∫log^n-1x dx

• ∫xⁿlog^mx dx = xⁿ⁺¹log^mx/ (n+1) – {(m)/(n+1)}.∫xⁿlog^m-1x dx

For algebraic functions, reduction formulas are:

• ∫ xⁿ/mxⁿ+k dx = x/m – y/k∫ 1/mxⁿ+k dx

For trigonometric functions, reduction formulas are:

• ∫ sinⁿx dx = -1/n sin^n-1x. cosx + (n-1_/n∫sin^n-2x dx

• ∫ cosⁿx dx = 1/n cos^n-1x.sinx + (n-1)/n∫cos^n-2x dx

• ∫ tanⁿx dx = 1/(n-1) tan^n-1x – ∫tan^n-2x dx

• ∫ sinⁿx.cos^mx dx = sinⁿ⁺¹x. cos^m-1x / (n+m) + (m-1)/(n+m)∫ sinⁿx.cos^m-2x dx

For exponential functions, reduction formulas are:

• ∫ xⁿe^mx dx = 1/m. xⁿe^mx – n/m ∫x^n-1e^mx dx

For inverse trigonometric functions, reduction formulas are:

• ∫ xⁿ arc sinx dx = (xⁿ⁺¹/n+1) arc sinx – (1/n+1)∫(xⁿ⁺¹/(1-x²)^1/2) dx

• ∫ xⁿ arc cosx dx = (xⁿ⁺¹/n+1) arc cosx + (1/n+1)∫(xⁿ⁺¹/(1-x²)^1/2) dx

• ∫ xⁿ arc tanx dx = (xⁿ⁺¹/n+1) arc tanx – (1/n+1)∫(xⁿ⁺¹/(1+x²)^1/2) dx

Article Related to Reduction Formula:

• Integration Formulas
• Integration by Parts
• Integration by Substitution Formula
• Integration by Partial Fractions

Example 1: Simplify ∫ x².log²x dx

Solution:

Using formula ∫xⁿlog^mx dx = xⁿ⁺¹log^mx/ n+1 – m/n+1 .∫xⁿlog^m-1x dx

n=2, m=2

∫ x².log²x dx = x³log²x/3 – 2/3.∫x²logx dx

= x³log²x/3 – 2/3.∫x²logx dx

= x³log²x/3 – 2/3. (x³.logx/3 – 1/3. ∫x² dx)

= x³log²x/3 – 2/3. (x³.logx/3 – 1/3. x³/3)

= x³log²x/3 – 2/9. x³.logx – 2/27. x³

Example 2: Simplify ∫ tan⁵x dx

Solution:  

Using formula ∫ tanⁿx dx = 1/n-1 tan^n-1x – ∫tan^n-2x dx

∫ tan⁵x dx = 1/4 tan⁴x – ∫tan³x dx

= 1/4 tan⁴x – ∫tan³x dx

= 1/4 tan⁴x – ( 1/2tan²x – ∫ tanx dx)

= 1/4 tan⁴x – 1/2tan²x  + 1/2. ln secx

Example 3: Simplify ∫ xe^3x dx

Solution:  

Using formula ∫ xⁿe^mx dx = 1/m. xⁿe^mx – n/m ∫x^n-1e^mx dx

= 1/3.xe^3x – n/m ∫e^3x dx

= 1/3.xe^3x – n/m . 3. e^3x dx

Example 4: Simplify ∫ log²x dx 

Solution:  

Using ∫ logⁿx dx = xlogⁿx -n∫log^n-1x dx

∫ log²x dx = 2log²x -2∫logx dx

= 2log²x -2∫logx dx

= 2log2x -2xlogx

Example 5: Simplify ∫ tan²x dx 

Solution: 

Using ∫ tanⁿx dx = 1/n-1 tan^n-1x – ∫tan^n-2x dx

n=2

∫ tan²x dx = tanx – ∫tan⁰x dx

∫ tan²x dx = tanx – x

What Is a Reduction Formula?

Reduction formula are formulas used in integration for solving integrals of higher order.

What is the Reduction Formula for Sin and Cos?

Integral Formula for sin and cos are:

• ∫ sinⁿx dx = -1/n sin^n-1x. cosx + (n-1_/n∫sin^n-2x dx

• ∫ cosⁿx dx = 1/n cos^n-1x.sinx + (n-1)/n∫cos^n-2x dx

Is Reduction Formula Important for JEE?

Reduction formula is very important for scoring good marks in JEE Main and Advanced as it helps solve integrals of higher order easily.