Calculus/Integration techniques/Reduction Formula

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Integration techniques/Reduction Formula

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^ne^xdx

Integration by parts allows us to simplify this to

I_n=x^ne^x-n\int x^{n-1}e^xdx=
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\sec^n(\theta)d\theta

then integration by parts lets us simplify this to

I_n=\sec^{n-2}(\theta)\tan(\theta)-(n-2)\int\sec^{n-2}(\theta)\tan^2(\theta)d\theta

Using the trigonometric identity, \tan^2(\theta)=\sec^2(\theta)-1 , we can now write

I_n =\sec^{n-2}(\theta)\tan(\theta)+(n-2)\left(\int\sec^{n-2}(\theta)d\theta-\int\sec^n(\theta)d\theta\right)
=\sec^{n-2}(\theta)\tan(\theta)+(n-2)\left(I_{n-2}-I_n\right)

Rearranging, we get

I_n=\frac{\sec^{n-2}(\theta)\tan(\theta)}{n-1}+\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.

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Integration techniques/Reduction Formula