Integration by reduction formulae

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Integration by reduction formula in integral calculus is a technique of integration, in the form of a recurrence relation. It is used when an expression containing an integer parameter, usually in the form of powers of elementary functions, or products of transcendental functions and polynomials of arbitrary degree, can't be integrated directly. But using other methods of integration a reduction formula can be set up to obtain the integral of the same or similar expression with a lower integer parameter, progressively simplifying the integral until it can be evaluated. [1] This method of integration is one of the earliest used.

How to find the reduction formula[edit]

The reduction formula can be derived using any of the common methods of integration, like integration by substitution, integration by parts, integration by trigonometric substitution, integration by partial fractions, etc. The main idea is to express an integral involving an integer parameter (e.g. power) of a function, represented by In, in terms of an integral that involves a lower value of the parameter (lower power) of that function, for example In-1 or In-2. This makes the reduction formula a type of recurrence relation. In other words, the reduction formula expresses the integral

I_n =\int f(x,n) \,dx,

in terms of

I_k =  \int f(x,k) \,dx,

where

k < n.

How to compute the integral[edit]

To compute the integral, we set n to its value and use the reduction formula to calculate the (n – 1) or (n – 2) integral. The higher index integral can be used to calculate lower index ones; the process is continued repeatedly until we reach a point where the function to be integrated can be computed, usually when its index is 0 or 1. Then we back-substitute the previous results until we have computed In. [2]

Examples[edit]

Below are examples of the procedure.

Cosine integral

Typically, integrals like

\int \cos^n x dx , \,\!

can be evaluated by a reduction formula.

\int \cos^n (x) dx\!, for n = 1, 2 ... 30

Start by setting:

I_n = \int \cos^n x dx . \,\!

Now re-write as:

I_n = \int \cos^ {n-1} x \cos x dx , \,\!

Integrating by this substitution:

\cos x dx = d ( \sin x) , \,\!
I_n = \int \cos^{n-1} x d(\sin x) . \!

Now integrating by parts:

 \begin{align} \int \cos^n x dx & = \cos^{n-1} x \sin x - \int \sin x d(\cos^{n-1} x) \\
& = \cos^{n-1} x \sin x + (n-1) \int \sin x \cos^{n-2} x\sin x dx\\
& = \cos^{n-1} x \sin x + (n-1) \int \cos^{n-2} x \sin^2 x dx\\
& = \cos^{n-1} x \sin x + (n-1) \int \cos^{n-2} x (1-\cos^2 x )dx\\
& = \cos^{n-1} x \sin x + (n-1) \int \cos^{n-2} x dx - (n-1)\int \cos^n x dx\\
& = \cos^{n-1} x \sin x + (n-1) I_{n-2} - (n-1) I_n ,
\end{align} \,

solving for In:

I_n \ + (n-1) I_n\ = \cos^{n-1} x \sin x\ + \ (n-1) I_{n-2} , \,
n I_n\ = \cos^{n-1} (x) \sin x\ + (n-1) I_{n-2} , \,
I_n \ = \frac{1}{n}\cos^{n-1} x \sin x\ + \frac{n-1}{n} I_{n-2} , \,

so the reduction formula is:

\int \cos^n x dx\ = \frac{1}{n}\cos^{n-1} x \sin x + \frac{n-1}{n} \int \cos^{n-2} x dx . \!

To supplement the example, the above can be used to evaluate the integral for (say) n = 5;

 I_5 = \int \cos^5 x dx . \,\!

Calculating lower indices:

n=5, \quad I_5 = \tfrac{1}{5} \cos^4 x \sin x + \tfrac{4}{5} I_3 , \,
n=3, \quad I_3 = \tfrac{1}{3} \cos^2 x \sin x + \tfrac{2}{3} I_1, \,

back-substituting:

\because I_1\ = \int \cos x dx = \sin x + C_1,\,
\therefore I_3\ = \tfrac{1}{3} \cos^2 x \sin x + \tfrac{2}{3}\sin x + C_2, \quad C_2\ = \tfrac{2}{3} C_1,\,
I_5\ = \frac{1}{5} \cos^4 x \sin x + \frac{4}{5}\left[\frac{1}{3} \cos^2 x \sin x + \frac{2}{3} \sin x\right] + C,\,

where C is a constant.

Exponential integral

Another typical example is:

\int x^n e^{ax} dx . \,\!.

Start by setting:

I_n = \int x^n e^{ax} dx . \,\!

Integrating by substitution:

 x^n dx = \frac{d ( x^{n+1})}{n+1} , \,\!
I_n = \frac{1}{n+1} \int e^{ax} d(x^{n+1}) , \!

Now integrating by parts:

\begin{align} \int e^{ax} d(x^{n+1}) & = x^{n+1}e^{ax} - \int x^{n+1} d(e^{ax}) \\
& = x^{n+1}e^{ax} - a \int x^{n+1} e^{ax}dx ,
\end{align} \!
(n+1) I_n = x^{n+1}e^{ax} - a I_{n+1} , \!

shifting indices back by 1 (so n + 1n, nn – 1):

n I_{n-1} = x^ne^{ax} - a I_n , \!

solving for n:

 I_n = \frac{1}{a} \left ( x^ne^{ax} - n I_{n-1} \right ) , \,\!

so the reduction formula is:

 \int x^n e^{ax} dx = \frac{1}{a} \left ( x^ne^{ax} - n \int x^{n-1} e^{ax} dx \right ). \!

Tables of integral reduction formulas[edit]

Rational functions[edit]

The following integrals[3] contain:

  • Factors of the linear radical \sqrt{ax+b}\,\!
  • Linear factors {px+q}\,\! and the linear radical \sqrt{ax+b}\,\!
  • Quadratic factors x^2+a^2\,\!
  • Quadratic factors x^2-a^2\,\!, for x>a\,\!
  • Quadratic factors a^2-x^2\,\!, for x<a\,\!
  • (Irreducible) quadratic factors ax^2+bx+c\,\!
  • Radicals of irreducible quadratic factors \sqrt{ax^2+bx+c}\,\!
Integral Reduction formula
I_n = \int \frac{x^n}{\sqrt{ax+b}} dx\,\! I_n = \frac{2x^n\sqrt{ax+b}}{a(2n+1)} - \frac{2nb}{a(2n+1)} I_{n-1}\,\!
I_n = \int \frac{dx}{x^n\sqrt{ax+b}}\,\! I_n = -\frac{\sqrt{ax+b}}{(n-1)bx^{n-1}}-\frac{a(2n-3)}{2b(n-1)}I_{n-1}\,\!
I_n = \int x^n\sqrt{ax+b}dx\,\! I_n = \frac{2x^n\sqrt{(ax+b)^3}}{a(2n+3)}-\frac{2nb}{a(2n+3)}I_{n-1}\,\!
I_n = \int \frac{\sqrt{ax+b}}{x^n}dx\,\! I_n = -\frac{\sqrt{ax+b}}{(n-1)x^{n-1}}+\frac{a}{2(n-1)}I_{n-1}\,\!
I_{n,m} = \int \frac{dx}{(ax+b)^n(px+q)^m}\,\! I_{n,m} = -\frac{1}{(n-1)(bp-aq)} \left [ \frac{1}{(ax+b)^{m-1}(px+q)^{n-1}}+a(n+m-2)I_{m,n-1} \right ]\,\!
I_{n,m} = \int  \frac{(ax+b)^m}{(px+q)^n} dx\,\! I_{n,m} =
\begin{cases}
  -\frac{1}{(n-1)(bp-aq)}\left [ \frac{(ax+b)^{m+1}}{(px+q)^{n-1}}+a(n+m-2)I_{m-1,n-1} \right ]   \\
  -\frac{1}{(n-m-1)p}\left [ \frac{(ax+b)^m}{(px+q)^{n-1}}+m(bp-aq)I_{m-1,n} \right ]  \\
  -\frac{1}{(n-1)p}\left [ \frac{(ax+b)^m}{(px+q)^{n-1}}-amI_{m-1,n-1} \right ]
\end{cases}\,\!
Integral Reduction formula
I_n=\int \frac{(px+q)^n}{\sqrt{ax+b}} dx\,\! \int (px+q)^n\sqrt{ax+b} dx  = \frac{2(px+q)^{n+1}\sqrt{ax+b}}{p(2n+3)}+\frac{bp-aq}{p(2n+3)}I_n\,\!

I_n=\frac{2(px+q)^n\sqrt{ax+b}}{a(2n+1)}+\frac{2n(aq-bp)}{a(2n+1)}I_{n-1}\,\!

I_n=\int \frac{dx}{(px+q)^n\sqrt{ax+b}}\,\! \int \frac{\sqrt{ax+b}}{(px+q)^n}dx = -\frac{\sqrt{ax+b}}{p(n-1)(px+q)^{n-1}}+\frac{a}{2p(n-1)}I_n\,\!

I_n= -\frac{\sqrt{ax+b}}{(n-1)(aq-bp)(px+q)^{n-1}}+\frac{a(2n-3)}{2(n-1)(aq-bp)}I_{n-1}\,\!

Integral Reduction formula
I_n= \int \frac{dx}{(x^2+a^2)^n}\,\! I_n= \frac{x}{2a^2(n-1)(x^2+a^2)^{n-1}}+\frac{2n-3}{2a^2(n-1)}I_{n-1}\,\!
I_{n,m}= \int \frac{dx}{x^m(x^2+a^2)^n}\,\! a^2I_{n,m}= I_{m,n-1}-I_{m-2,n}\,\!
I_{n,m}= \int \frac{x^m}{(x^2+a^2)^n} dx\,\! I_{n,m}= I_{m-2,n-1}-a^2I_{m-2,n}\,\!
Integral Reduction formula
I_n= \int \frac{dx}{(x^2-a^2)^n}\,\! I_n= -\frac{x}{2a^2(n-1)(x^2-a^2)^{n-1}}-\frac{2n-3}{2a^2(n-1)}I_{n-1}\,\!
I_{n,m}= \int \frac{dx}{x^m(x^2-a^2)^n}\,\! {a^2}I_{n,m}= I_{m-2,n}-I_{m,n-1}\,\!
I_{n,m}= \int \frac{x^m}{(x^2-a^2)^n} dx\,\! I_{n,m}= I_{m-2,n-1}+a^2I_{m-2,n}\,\!
Integral Reduction formula
I_n= \int \frac{dx}{(a^2-x^2)^n}\,\! I_n= \frac{x}{2a^2(n-1)(a^2-x^2)^{n-1}}+\frac{2n-3}{2a^2(n-1)}I_{n-1}\,\!
I_{n,m}= \int \frac{dx}{x^m(a^2-x^2)^n}\,\! {a^2}I_{n,m}= I_{m,n-1}+I_{m-2,n}\,\!
I_{n,m}= \int \frac{x^m}{(a^2-x^2)^n} dx\,\! I_{n,m}= a^2I_{m-2,n}-I_{m-2,n-1}\,\!
Integral Reduction formula
I_n = \int \frac{dx}{{x^n}(ax^2+bx+c)}\,\! -cI_n =\frac{1}{x^{n-1}(n-1)}+ bI_{n-1}+aI_{n-2}\,\!
I_{m,n}=\int \frac{x^m dx}{(ax^2+bx+c)^n}\,\! I_{m,n}= -\frac{x^{m-1}}{a(2n-m-1)(ax^2+bx+c)^{n-1}} - \frac{b(n-m)}{a(2n-m-1)}I_{m-1,n} + \frac{c(m-1)}{a(2n-m-1)}I_{m-2,n}\,\!
I_{m,n}= \int \frac{dx}{x^m(ax^2+bx+c)^n}\,\! -c(m-1)I_{m,n}= \frac{1}{x^{m-1}(ax^2+bx+c)^{n-1}}+{a(m+2n-3)}I_{m-2,n}+{b(m+n-2)}I_{m-1,n}\,\!
Integral Reduction formula
I_n = \int (ax^2+bx+c)^ndx\,\! 8a(n+1)I_{n+\frac{1}{2}} = 2(2ax+b)(ax^2+bx+c)^{n+\frac{1}{2}} + (2n+1)(4ac-b^2)I_{n-\frac{1}{2}}\,\!
I_n = \int \frac{1}{(ax^2+bx+c)^n}dx\,\! (2n-1)(4ac-b^2)I_{n+\frac{1}{2}} = \frac{2(2ax+b)}{(ax^2+bx+c)^{n-\frac{1}{2}}}+{8a(n-1)}I_{n-\frac{1}{2}}\,\!

note that by the laws of indices:

I_{n+\frac{1}{2}} =  I_{\frac{2n+1}{2}} =\int \frac{1}{(ax^2+bx+c)^{\frac{2n+1}{2}}}dx = \int \frac{1}{\sqrt{(ax^2+bx+c)^{2n+1}}}dx\,\!

Transcendental functions[edit]

See main article: Transcendental function

The following integrals[4] contain:

  • Factors of sine
  • Factors of cosine
  • Factors of sine and cosine products and quotients
  • Products/quotients of exponential factors and powers of x
  • Products of exponential and sine/cosine factors
Integral Reduction formula
I_n=\int x^n \sin{ax} dx\,\! a^2I_n=-ax^n \cos{ax} + nx^{n-1} \sin{ax} - n(n-1) I_{n-2} \,\!
J_n=\int x^n \cos{ax} dx \,\! a^2J_n=ax^n \sin{ax} + nx^{n-1} \cos{ax} - n(n-1) J_{n-2} \,\!
 I_n = \int \frac{\sin{ax}}{x^n} dx\,\!

J_n = \int \frac{\cos{ax}}{x^n} dx \,\!

I_n = -\frac{\sin{ax}}{(n-1)x^{n-1}}+\frac{a}{n-1}J_{n-1}\,\!

J_n = -\frac{\cos{ax}}{(n-1)x^{n-1}}-\frac{a}{n-1}I_{n-1}\,\!

the formulae can be combined to obtain separate equations in In:

J_{n-1} = -\frac{\cos{ax}}{(n-2)x^{n-2}}-\frac{a}{n-2}I_{n-2}\,\!

I_n = -\frac{\sin{ax}}{(n-1)x^{n-1}}-\frac{a}{n-1}\left [\frac{\cos{ax}}{(n-2)x^{n-2}}+\frac{a}{n-2}I_{n-2}\right ] \,\!

 \therefore I_n = -\frac{\sin{ax}}{(n-1)x^{n-1}}-\frac{a}{(n-1)(n-2)}\left (\frac{\cos{ax}}{x^{n-2}}+aI_{n-2}\right ) \,\!

and Jn:

I_{n-1} = -\frac{\sin{ax}}{(n-2)x^{n-2}}+\frac{a}{n-2}J_{n-2}\,\!

J_n = -\frac{\cos{ax}}{(n-1)x^{n-1}}-\frac{a}{n-1}\left [-\frac{\sin{ax}}{(n-2)x^{n-2}}+\frac{a}{n-2}J_{n-2}  \right ]\,\!

 \therefore J_n = -\frac{\cos{ax}}{(n-1)x^{n-1}}-\frac{a}{(n-1)(n-2)}\left (-\frac{\sin{ax}}{x^{n-2}}+aJ_{n-2}  \right )\,\!

I_n = \int \sin^n{ax} dx\,\! anI_n = -\sin^{n-1}{ax}\cos{ax}+a(n-1)I_{n-2}\,\!
J_n = \int \cos^n{ax} dx\,\! anJ_n = \sin{ax}\cos^{n-1}{ax}+a(n-1)J_{n-2}\,\!
I_n = \int \frac{dx}{\sin^n{ax}}\,\! (n-1)I_n = - \frac{\cos{ax}}{a\sin^{n-1}{ax}}+ (n-2)I_{n-2}\,\!
J_n = \int \frac{dx}{\cos^n{ax}}\,\! (n-1)J_n = \frac{\sin{ax}}{a\cos^{n-1}{ax}}+ (n-2)I_{n-2}\,\!
Integral Reduction formula
I_{m,n} = \int \sin^m{ax}\cos^n{ax}dx\,\! I_{m,n} = \begin{cases}
    -\frac{\sin^{m-1}{ax}\cos^{n+1}{ax}}{a(m+n)}+\frac{m-1}{m+n}I_{m-2,n} \\
    \frac{\sin^{m+1}{ax}\cos^{n-1}{ax}}{a(m+n)}+\frac{n-1}{m+n}I_{m,n-2} \\
\end{cases}\,\!
I_{m,n} = \int \frac{dx}{\sin^m{ax}\cos^n{ax}}\,\! I_{m,n} = \begin{cases}
    \frac{1}{a(n-1)\sin^{m-1}{ax}\cos^{n-1}{ax}}+\frac{m+n-2}{n-1}I_{m,n-2} \\
    -\frac{1}{a(m-1)\sin^{m-1}{ax}\cos^{n-1}{ax}}+\frac{m+n-2}{m-1}I_{m-2,n} \\
\end{cases}\,\!
I_{m,n} = \int \frac{\sin^m{ax}}{\cos^n{ax}}dx\,\! I_{m,n} = \begin{cases}
    \frac{\sin^{m-1}{ax}}{a(n-1)\cos^{n-1}{ax}}-\frac{m-1}{n-1}I_{m-2,n-2} \\
    \frac{\sin^{m+1}{ax}}{a(n-1)\cos^{n-1}{ax}}-\frac{m-n+2}{n-1}I_{m,n-2} \\
    -\frac{\sin^{m-1}{ax}}{a(m-n)\cos^{n-1}{ax}}+\frac{m-1}{m-n}I_{m-2,n} \\
\end{cases}\,\!
I_{m,n} =  \int \frac{\cos^m{ax}}{\sin^n{ax}}dx\,\! I_{m,n} = \begin{cases}
    -\frac{\cos^{m-1}{ax}}{a(n-1)\sin^{n-1}{ax}}-\frac{m-1}{n-1}I_{m-2,n-2} \\
    -\frac{\cos^{m+1}{ax}}{a(n-1)\sin^{n-1}{ax}}-\frac{m-n+2}{n-1}I_{m,n-2} \\
    \frac{\cos^{m-1}{ax}}{a(m-n)\sin^{n-1}{ax}}+\frac{m-1}{m-n}I_{m-2,n} \\
\end{cases}\,\!
Integral Reduction formula
I_{n} = \int x^n e^{ax} dx\,\!

n > 0\,\!

 I_{n} = \frac{x^n e^{ax}}{a} - \frac{n}{a}I_{n-1} \,\!
I_{n} = \int x^{-n} e^{ax} dx\,\!

n > 0\,\!

n \neq 1\,\!

 I_{n} = -\frac{- e^{ax}}{(n-1)x^{n-1}} - \frac{a}{n-1}I_{n-1} \,\!
I_{n} = \int x^{ax} \sin^n{bx} dx\,\!  I_{n} = -\frac{x^{ax} \sin^{n-1}{bx}}{a^2+(bn)^2}\left ( a\sin bx - bn\cos bx \right ) + \frac{n(n-1)b^2}{a^2+(bn)^2}I_{n-2} \,\!
I_{n} = \int x^{ax} \cos^n{bx} dx\,\!  I_{n} = -\frac{x^{ax} \cos^{n-1}{bx}}{a^2+(bn)^2}\left ( a\cos bx + bn\sin bx \right ) + \frac{n(n-1)b^2}{a^2+(bn)^2}I_{n-2} \,\!

References[edit]

  1. ^ Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3
  2. ^ Further Elementary Analysis, R.I. Porter, G. Bell & Sons Ltd, 1978, ISBN 0-7135-1594-5
  3. ^ http://www.sosmath.com/tables/tables.html -> Indefinite integrals list
  4. ^ http://www.sosmath.com/tables/tables.html -> Indefinite integrals list

Bibliography[edit]

  • Anton, Bivens, Davis, Calculus, 7th edition.