Reduction Formula

Reduction Formula

Integration in an important part of calculus. Reduction is one of method used in integration. Let us know more about it.

Definition

The reduction formula is an essential method of integration, in order to solve higher order integrals. Solving the higher order/degree integrals with simple integration can be very tedious and time consuming, so to help decrease the time and to increase the possibility of solving the problem, reduction formula can be applied to it. The reduction formula is derived from the basic integration formula and same rules are applied in it.

Formula

The following formulas can be helpful while working with higher order problems like algebraic variables, logarithmic functions, and trigonometric functions.

Formula 1

For an Exponential Expression:

∫xⁿ.e^mx.dx = (1/m).xⁿ.e^mx – (n/m) ∫x^n-1.e^mx.dx

Formula 2

For logarithmic expressions:

• ∫x .dx = xlogⁿ x – n∫x .dx

• ∫xⁿx .dx = (x /n+1) – (m/n+1) ∫xⁿlog^m-1x.dx

Formula 3 

For trigonometric Functions:

• ∫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

• ∫Sinⁿx.Cos^mx.dx = (Sinⁿ⁺¹x.Cos^m-1x/n+m) + (m-1/n+m)∫Sinⁿx.Cos^m-2x.dx

• ∫Tanⁿx.dx = (1/n-1).Tan^n-1x – ∫Tan^n-2 x.dx

Formula 4

For algebraic expressions:

∫(xⁿ/aⁿ+b)  .dx = (x/a) – (b/a) ∫(1/axⁿ+b) .dx

Solved examples:

Let us try to solve a few questions:

1. Find the integral of Sin⁶x.

Solution: 

Reduction formula used here is:

∫Sinⁿx.dx = (1/n) Sin^n-1x.Cosx + (n-1/n)∫Sin^n-2x.dx

∫Sin⁶x.dx = (-1/6).Sin⁵x.Cosx + (5/6) ∫Sin⁴x.dx

∫Sin⁶x.dx = (-1/6).Sin⁵x.Cosx + (5/6) (∫(4Sinx-Sin4x)/5).dx)

= (1/6).Sin⁵x.Cosx + (1/6)(4∫Sinx.dx – ∫Sin4x.dx)

= (1/6).Sin⁵x.Cosx + (1/6)(-4Cosx + (Cos4x/4))

∴∫Sin⁶x.dx = (1/6).Sin⁵x.Cosx + (4Cosx/6) + (Cos4x/24)

2. Find the integral of log2 x.

With help of the following equation, we can find that:

∫x .dx = xlogⁿ x – n∫x .dx

∫x .dx = 2x – 2log x.dx 

∫x .dx = 2x – 2xlogx-x

∫x .dx = 2x – 2xlogx+2x

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