Reduction Formula

Reduction Formula is a powerful technique used in integration to simplify complex integrals by expressing them in terms of lower-order or simple integrals.

This method is especially useful when dealing by expressing them of lower-order or simple integrals. This method is especially useful when dealing with integrals involving:

• Higher powers of elementary functions
• Polynomial of arbitrary degree.
• Functions that do not have a direct or simple antiderivative.

Reduction formulas for different expressions are listed below:

Reduction Formulas for Logarithmic Functions

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

Reduction Formulas for Algebraic Functions

For algebraic functions, reduction formulas are:

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

Reduction Formulas for Trigonometric Functions

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

Reduction Formulas for Exponential Functions

For exponential functions, reduction formulas are:

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

Reduction Formulas for Inverse Trigonometric Functions

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

Related Articles:

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

Examples Using Reduction Formula

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

Unsolved Questions on Reduction Formula

Question 1: Simplify using the reduction formula: ∫x³(ln⁡x)² dx

Question 2: Evaluate using reduction formula: ∫sin⁡5x dx

Question 3: Using the reduction method, find: ∫x²e^4x dx

Question 4: Simplify using reduction formula: ∫sec⁡⁴xtan⁡³x dx

Question 5: Simplify using the log-power reduction formula: ∫(ln⁡x)³ dx

Question 6: Find the integral using reduction: ∫tan⁡⁶x dx