Reduction Formula
A reduction formula is often used in integration for working out integrals of higher order. It is lengthy and tedious to work across higher degree expressions, and here the reduction formulas are given as simple expressions with a degree n, to solve these higher degree expressions. These reduction formulas have been derived from the base formulas of integration and work with the same rules of integration.
What is the Reduction Formula?
The following reduction formulas are helpful to work across higher degree expressions involving algebraic variables, trigonometry functions, logarithmic functions. The reduction formulas have been presented below as a set of four formulas.
Formula 1
Reduction Formula for basic exponential expressions.
\(\int x^n.e^{mx}.dx = \frac{1}{m}. x^n.e^{mx}  \frac{n}{m}\int x^{n1}.e^{mx}.dx\)
Formula 2
Reduction Formula for logarithmic expressions.

\(\int log^nx.dx = xlog^nx n\int log^{n  1}x.dx \)

\(\int x^nlog^mx.dx = \frac{x^{n + 1}log^mx}{n + 1}  \frac{m}{n + 1}\int x^n log^{m  1}x.dx\)
Formula 3
Reduction Formula for trigonometric functions.

\(\int Sin^nx.dx = \frac{1}{n}Sin^{n  1}x.Cosx +\frac{n  1}{n }\int Sin^{ n 2}x.dx\)

\(\int Cos^nx.dx = \frac{1}{n}Cos^{n  1}x.Sinx +\frac{n  1}{n }\int Cos^{ n 2}x.dx\)

\(\int Sin^nx.Cos^mx.dx = \frac{Sin^{n + 1}x.Cos^{m  1}x}{n + m} +\frac{m  1}{n + m}\int Sin^nx.Cos^{ m 2}x.dx\)

\(\int Tan^nx.dx = \frac{1}{n  1}.Tan^{n  1}x \int Tan^{n  2}x.dx\)
Formula 4
Reduction Formula for algebraic expressions.
\(\int \frac{x^n}{ax^n +b}.dx = \frac{x}{a} \frac{b}{a}\int \frac{1}{ax^n + b}.dx\)
Let us try out a few examples to more clearly understand, how to use the reduction formula.
Examples Using Reduction Formula
Example 1: Find the integral of Sin^{5}x.
Solution:
We can apply the reduction formula to find the integral of Sin^{5}x.
\(\begin{align}\int Sin^nx.dx &= \frac{1}{n}Sin^{n  1}x.Cosx +\frac{n  1}{n }\int Sin^{ n 2}x.dx\\\int Sin^5x.dx&=\frac{1}{5}.Sin^4x.Cosx + \frac{4}{5}\int Sin^3 x.dx \\&=\frac{1}{5}.Sin^4x.Cosx + \frac{4}{5}(\int \frac{(3sinx  sin3x)}{4}.dx)\\&=\frac{1}{5}.Sin^4x.Cosx + \frac{1}{5}(3\int Sinx.dx  \int Sin3x.dx)\\&=\frac{1}{5}.Sin^4x.Cosx + \frac{1}{5}(3Cosx +\frac{Cos3x}{3})\\&=\frac{1}{5}.Sin^4x.Cosx \frac{3Cosx}{5} +\frac{Cos3x}{15}\end{align}\)
Answer: \(\int Sin^5x.dx =\dfrac{1}{5}.Sin^4x.Cosx \dfrac{3Cosx}{5} +\dfrac{Cos3x}{15}\)
Example 2: Evaluate the integral of x^{3}Log^{2}x.
Solution:
Applying the reduction formula we can conveniently find the integral of the given expression.
\(\begin{align}\int x^nlog^mx.dx &= \frac{x^{n + 1}log^mx}{n + 1}  \frac{m}{n + 1}\int x^n log^{m  1}x.dx
\\ \int x^3log^2x.dx &= \frac{x^4log^2x}{4}  \frac{2}{4}\int x^ 3logx.dx\\&= \frac{x^4log^2x}{4}  \frac{1}{2}(\frac{x^4logx}{4}  \frac{1}{4}.\int x^3.dx)\\&= \frac{x^4log^2x}{4} \frac{x^4logx}{8} + \frac{x^4}{32} + C\end{align}\)
Answer: \( \int x^3log^2x.dx = \dfrac{x^4log^2x}{4} \dfrac{x^4logx}{8} + \dfrac{x^4}{32} + C\)
Example 3: Evaluate: \(\int \tan ^{5}(2 x) d x\)
Solution:
Use: \(\operatorname{stan}^{n}(u) d u=(1 / n1) \tan ^{n1}(u)\int \tan ^{n2}(u) d u\)
Substitution:
\(\mathrm{a}=2 \mathrm{x}\)
therefore \(1 / 2 \mathrm{da}=\mathrm{dx}\)
Hence,
\(\int \tan ^{5}(2 x) d x=1 / 2\left[\left[\tan ^{5}(a) d a\right]\right.\)
\(=1 / 2\left[1 / 4 \tan ^{4}(a)\int \tan ^{3}(a) d a\right]\)
\(=1 / 2\left[1 / 4 \tan ^{4}(\mathrm{a})\left[\frac{1}{2} \tan ^{2}(\mathrm{a})\int \tan (\mathrm{a}) \mathrm{da}\right]\right]\)
\(=1 / 8 \tan ^{4}(\mathrm{a})1 / 4 \tan ^{2}(\mathrm{a})+1 / 2 \ln \sec (\mathrm{a})+\mathrm{C}\)
\(=1 / 8 \tan ^{4}(2 x)1 / 4 \tan ^{2}(2 x)+1 / 2 \ln \sec (2 x)+C\)
Answer: \(\int \tan ^{5}(2 x) d x =1 / 8 \tan ^{4}(2 x)1 / 4 \tan ^{2}(2 x)+1 / 2 \ln \sec (2 x)+C\)
FAQs on Reduction Formula
What Is a Reduction Formula?
A reduction formula is often used in integration for working out integrals of higher order. It is lengthy and tedious to work across higher degree expressions, and here the reduction formulas are given as simple expressions with a degree n, to solve these higher degree expressions. The following reduction formulas are helpful to work across higher degree expressions involving algebraic variables, trigonometry functions, logarithmic functions. The reduction formulas have been presented here as a set of four formulas.
What Is the Purpose of the Reduction Formula?
The reduction formula is used when the given integral cannot be evaluated otherwise. The repeated application of the reduction formula helps us to evaluate the given integral.
What Is the Reduction Formula in Trigonometry?
The following reduction formulas are helpful to work across higher degree expressions involving algebraic variables, trigonometry functions, logarithmic functions. Any trigonometric function whose argument is 90°±θ, 180°±θ, 270°±θ and 360°±θ (hence θ) can be written simply in terms of θ. For example, you may have noticed that the cosine graph is identical to the sine graph except for a phase shift of 90°. From this, we may expect that sin(90°+θ)=cosθ.
What Are the Reduction Formulas for Trigonometric Functions?
Reduction Formula for trigonometric functions are:
 \(\int Sin^nx.dx = \frac{1}{n}Sin^{n  1}x.Cosx +\frac{n  1}{n }\int Sin^{ n 2}x.dx\)
 \(\int Cos^nx.dx = \frac{1}{n}Cos^{n  1}x.Sinx +\frac{n  1}{n }\int Cos^{ n 2}x.dx\)
 \(\int Sin^nx.Cos^mx.dx = \frac{Sin^{n + 1}x.Cos^{m  1}x}{n + m} +\frac{m  1}{n + m}\int Sin^nx.Cos^{ m 2}x.dx\)
 \(\int Tan^nx.dx = \frac{1}{n  1}.Tan^{n  1}x \int Tan^{n  2}x.dx\)
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