Skip to main content

10. Reduction Formulae

by M. Bourne

You may have noticed in the Table of Integrals that some integrals are given in terms of a simpler integral. These require a few steps to find the final answer.

Reduction formulae are integrals involving some variable `n`, as well as the usual `x`. They are normally obtained from using integration by parts.

We use the notation `I_n` when writing reduction formulae.

Example 1

Given the reduction formula

`I_n=intsin^nx\ dx` `=-1/ncos x\ sin^(n-1)x+(n-1)/nI_(n-2),`

find `intsin^4x\ dx`.


Applying the reduction formula with `n = 4` gives:

`int sin^4x\ dx` `=-1/4cos x\ sin^3x+3/4I_2`

We now need to find `I_2=intsin^2x\ dx`, which corresponds to `n=2`.

Now, from our table of integrals,

`intsin^2xdx` `=x/2-1/2sin x\ cos x+K`

So putting it all together gives:

`intsin^4x\ dx`

`=-1/4cos x\ sin^3x` `+3/4[x/2-1/2sin x\ cos x+K]`

`=-1/4cos x\ sin^3x` `+3/8x` ` -3/8 cos x\ sin x+K^'`

`=3/8 x - 1/4 sin(2 x)` `+ 1/32 sin(4 x)+K^'` (afer some manipulation)

NOTE: We used `K` and `K^'` since the value of those constants is actually different.

Easy to understand math videos:

For interest, here are the graphs of the function `y=sin^4 x` and its integral.

Graph of y(x)=sin^4(x) - question involves reduction formulae Graph of integral by reduction formulae

Graph of `y(x)=sin^4 x`, and its integral `y=3/8 x - 1/4 sin(2 x)+ 1/32 sin(4 x)` (I've used `K=0`).

Example 2

We are given that if

`I_n=inttan^nx\ dx`,

then `I_n=1/(n-1)tan^(n-1)x-I_(n-2).`

Find `inttan^3x\ dx`.


`inttan^3x\ dx=1/2tan^2x-I_1`

Now `I_1=inttan x\ dx` `=-ln\ |cos x|+K`

So `inttan^3x\ dx` `=1/2tan^2x+ln\ |cos x|+K`

Note: Often in this work there is more than one way to attack the question, so it is possible to get a correct answer, but in a different form. Using math software, like Scientific Notebook, you may get:

`inttan^3x\ dx` `=1/2tan^2x-1/2ln\ |1+tan^2x|`

Are the 2 answers the same?

Please support IntMath!


Search IntMath, blog and Forum

Search IntMath

Online Calculus Solver

This calculus solver can solve a wide range of math problems.

Calculus Lessons on DVD

Math videos by

Easy to understand calculus lessons on DVD. See samples before you commit.

More info: Calculus videos

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

See the Interactive Mathematics spam guarantee.