Skip to main content
Search IntMath
Close

450+ Math Lessons written by Math Professors and Teachers

5 Million+ Students Helped Each Year

1200+ Articles Written by Math Educators and Enthusiasts

Simplifying and Teaching Math for Over 23 Years

Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class.

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`.

Answer

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.

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`.

Answer

`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?

Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class.