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^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 \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

\begin{matrix}
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) \\
\end{matrix}

Rearranging, we get

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

← Integration techniques/Tangent Half Angle Calculus Integration techniques/Irrational Functions →
Integration techniques/Reduction Formula