Reduction Formula (original) (raw)

Last Updated : 7 Nov, 2025

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:

• **∫ log n x dx = xlog n x -n∫log n-1 x dx
• **∫x n log m x dx = x n+1 log m x/ (n+1) - {(m)/(n+1)}.∫x n log m-1 x dx

**Reduction Formulas for Algebraic Functions

For algebraic functions, reduction formulas are:

• **∫ x n /mx n +k dx = x/m - y/k∫ 1/mx n +k dx

**Reduction Formulas for Trigonometric Functions

For trigonometric functions, reduction formulas are:

• **∫ sin n x dx = -1/n sin n-1 x. cosx + (n-1_/n∫sin n-2 x dx
• **∫ cos n x dx = 1/n cos n-1 x.sinx + (n-1)/n∫cos n-2 x dx
• **∫ tan n x dx = 1/(n-1) tan n-1 x - ∫tan n-2 x dx
• **∫ sin n x.cos m x dx = sin n+1 x. cos m-1 x / (n+m) + (m-1)/(n+m)∫ sin n x.cos m-2 x dx

**Reduction Formulas for Exponential Functions

For exponential functions, reduction formulas are:

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

**Reduction Formulas for Inverse Trigonometric Functions

For inverse trigonometric functions, reduction formulas are:

**∫ x n arc sinx dx = (x n+1 /n+1) arc sinx - (1/n+1)∫(x n+1 /(1-x 2 ) 1/2 ) dx

• **∫ x n arc cosx dx = (x n+1 /n+1) arc cosx + (1/n+1)∫(x n+1 /(1-x 2 ) 1/2 ) dx
• **∫ x n arc tanx dx = (x n+1 /n+1) arc tanx - (1/n+1)∫(x n+1 /(1+x 2 ) 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 ∫xnlogmx dx = xn+1logmx/ n+1 - m/n+1 .∫xnlogm-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 tann-1x - ∫tann-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 ∫ xe3x dx

**Solution:

Using formula ∫ xnemx dx = 1/m. xnemx - n/m ∫xn-1emx dx

= 1/3.xe3x - n/m ∫e3x dx

= 1/3.xe3x - n/m . 3. e3x dx

**Example 4: Simplify ∫ log2x dx

**Solution:

Using ∫ lognx dx = xlognx -n∫logn-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 tann-1x - ∫tann-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: ∫x2e4x 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