Reduction Formula

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.

The reduction formula comes to the rescue to simplify the higher-order terms. Integration of higher-order terms consisting of logarithmic, algebraic, and trigonometric functions are simplified by reduction formulas. n 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:

Reduction Formulas for Logarithmic expressions

∫ 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 expressions

∫ xⁿ/mxⁿ+k dx = x/m – y/k∫ 1/mxⁿ+k dx

Reduction Formulas for Trigonometric expressions

∫ 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 expressions

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

Reduction Formulas for Reduction Formulas for Inverse Trigonometric expressions 

∫ 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

Sample Problems

Problem1: 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

Problem2: 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

Problem3: 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

Problem4: 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

Problem5: 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