Further Maths (WJEC)

 

FP3 Integration by REDUCTION FORMULAE

Page history last edited by Steph Richards 6 mos ago

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

 

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