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?

\(\int x^n.e^{mx}.dx = \frac{1}{m}. x^n.e^{mx} - \frac{n}{m}\int x^{n-1}.e^{mx}.dx\)

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 Sin⁵x.

Solution:
We can apply the reduction formula to find the integral of Sin⁵x.
\(\begin{align}\int Sin^nx.dx &= -\frac{1}{n}Sin^{n - 1}x.Cosx +\frac{n - 1}{n }\int Sin^{ n- 2}x.dx\\\int Sin^5x.dx&=-\frac{1}{5}.Sin^4x.Cosx + \frac{4}{5}\int Sin^3 x.dx \\&=-\frac{1}{5}.Sin^4x.Cosx + \frac{4}{5}(\int \frac{(3sinx - sin3x)}{4}.dx)\\&=-\frac{1}{5}.Sin^4x.Cosx + \frac{1}{5}(3\int Sinx.dx - \int Sin3x.dx)\\&=-\frac{1}{5}.Sin^4x.Cosx + \frac{1}{5}(-3Cosx +\frac{Cos3x}{3})\\&=-\frac{1}{5}.Sin^4x.Cosx  -\frac{3Cosx}{5} +\frac{Cos3x}{15}\end{align}\) 

Answer: \(\int Sin^5x.dx =-\dfrac{1}{5}.Sin^4x.Cosx  -\dfrac{3Cosx}{5} +\dfrac{Cos3x}{15}\)

Example 2: Evaluate the integral of x³Log²x.

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: \(\operatorname{stan}^{n}(u) d u=(1 / n-1) \tan ^{n-1}(u)-\int \tan ^{n-2}(u) d u\)
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: \(\int \tan ^{5}(2 x) d x =1 / 8 \tan ^{4}(2 x)-1 / 4 \tan ^{2}(2 x)+1 / 2 \ln \sec (2 x)+C\)