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FP3 Integration by REDUCTION FORMULAE

Page history last edited by whitecorp 13 years ago Saved with comment

Integration by Parts is used to Generate Reduction Formulae that allows some quite Complex Integration to be carried out.

 

THESE NOTES ARE TAKEN FROM WIKIBOOKS

A reduction formula is one that enables us to solve an integral problem by reducing it to a problem of solving an easier integral problem, and then reducing that to the problem of solving an easier problem, and so on.

For example, if we let

I_n=\int x^n e^x\,dx

Integration by parts allows us to simplify this to

I_n=x^ne^x - n\int x^{n-1}e^x\,dx=
I_n=x^ne^x - nI_{n-1} \,\!

which is our desired reduction formula. Note that we stop at

I_0=e^x \,\!.

Similarly, if we let

I_n=\int_0^\alpha \sec^n \theta \, d\theta

then integration by parts lets us simplify this to

I_n=\sec^{n-2}\alpha \tan \alpha - (n-2)\int_0^\alpha \sec^{n-2} \theta \tan^2 \theta \, d\theta

Using the trigonometric identity, tan2=sec2-1, we can now write

\begin{matrix} I_n &=& \sec^{n-2}\alpha \tan \alpha & + (n-2) \left( \int_0^\alpha \sec^{n-2} \theta \, d\theta - \int_0^\alpha \sec^n \theta \, d\theta \right) \ &=& \sec^{n-2}\alpha \tan \alpha & + (n-2) \left( I_{n-2} - I_n \right) \ \end{matrix}

Rearranging, we get

I_n=\frac{1}{n-1}\sec^{n-2}\alpha \tan \alpha + \frac{n-2}{n-1} I_{n-2}

Note that we stop at n=1 or 2 if n is odd or even respectively.

As in these two examples, integrating by parts when the integrand contains a power often results in a reduction formula.

Retrieved from "http://en.wikibooks.org/wiki/Calculus/Integration_techniques/Reduction_Formula"

 

Watch a video by MIDNIGHTTUTOR on Repeated Integration by Parts

 

Integration By PARTS leads to a REDUCTION FORMULA

 

"JUST THE MATHS" REDUCTION FORMULAE NOTES

 

See also FP3 MIXED INTEGRATION METHODS

 

A explanatory piece on reduction formula together with many detailed worked problems.

 

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