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
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.
Table of Content
- What is Reduction Formula?
- Reduction Formulas for Logarithmic Functions
- Reduction Formulas for Algebraic Functions
- Reduction Formulas for Trigonometric Functions
- Reduction Formulas for Exponential Functions
- Reduction Formulas for Inverse Trigonometric Functions
- Examples Using Reduction Formula
- FAQs on Reduction Formula