Reduction Formulas For Integration

 how can we find the indefinite integral

of sine squared of X DX using the

reduction formula for sine now here's

the formula that you need to use the

integral of sine raised to the N of X DX

it's negative 1 over N cosine X sine to

the N minus 1 power of X plus n minus 1

over n times the integral of sine raised

to the N minus 2 power of X DX

so in this problem we can see that n is

equal to 2 so using the formula it's

gonna be negative 1 over N or negative 1

over 2 cosine X sine n minus 1 or 2

minus 1 which is 1 so we could simply

say sine X plus n minus 1 over n so

that's 2 minus 1 1 over 2 which is 1

over 2 integral sine n minus 2 so 2

minus 2 is 0

anything raised to the 0 power is 1 so

this becomes 1 DX now the integral of 1

DX is simply X so we get this answer

this is gonna be 1/2 X plus C now you

can leave your answer like this or you

can adjust it if you wish to adjust it

you need to be familiar with the double

angle formula for sine sine 2 X is 2

sine X cosine X so if we divide by 2 1/2

sine X

I mean 1/2 sine 2 X is sine x times

cosine X so what we're gonna do is we're

gonna replace sine X cosine X with 1/2

sine 2x so this is gonna be negative 1/2

times 1/2 sine 2x

plus half of X plus C and so I'm gonna

write the final answer like this so it's

gonna be 1/2 X and then the 1/2 times

1/2 is 1/4 but there's a negative sign

in front of it so it's negative 1/4 sine

2 X plus C so this is the antiderivative

of sine squared X you can write your

answer like that or you could write it

like this if you want to let's try

another example what is the

antiderivative of cosine cubed of X DX

using the reduction formula for cosine

so let's start with the formula cosine

raised to the end of X DX it's equal to

1 over n cosine to the N minus 1 of X

times sine X plus n minus 1 over N

integral cosine n minus 2 of X DX so in

this problem n is equal to 3 so this is

gonna equal 1 over N or 1 over 3 cosine

n minus 1 3 minus 1 is 2 and then times

sine X and then we have n minus 1 over N

so 3 minus 1 over 3 that's gonna be 2

over 3 integral cosine n minus 2 3 minus

2 is 1 and so we have 1/3

now I'm gonna replace cosine squared

with 1 minus sine squared ma I'll do

that later

but for now we have cosine squared times

sine X and the integral of cosine is

sine so if you want to you can leave

your answer like this but what I'm gonna

do is I'm gonna adjust it I'm gonna

replace cosine squared with 1 minus sine

squared

and then I'm gonna distribute 1/3 sine X

to 1 minus sine squared so it's gonna be

1/3 times 1 times sine X so that's

simply 1/3 sine X and then it's 1/3

times negative sine squared times sine X

which is a negative 1/3 sine to the

third power of X and then we have

everything else so now let's combine

like terms we can combine these two and

1 over 3 plus 2 over 3 is 3 over 3 and 3

divided by 3 is 1 so the final answer is

gonna be sine X minus 1/3 sine cube X

plus C and so this is the indefinite

integral of cosine cube X DX

so you can leave your answer like this

or you can write it in this form if you

want to now let's try one more problem

let's integrate sine to the fourth X DX

using the reduction formula for sine so

first let's rewrite the formula this is

what we had at the beginning it was a

negative 1 over N cosine and then times

sine n minus 1 plus n minus 1 over an

integral sign and minus 2 X DX so we can

clearly see that n is 4 in his prom so

it's gonna be negative 1 over N or

negative 1 over 4 cosine X and then sine

n minus 1 or 4 minus 1 which is 3 and

then n minus 1 that's 4 minus 1 again

that's 3 over N which is 4 integral sign

and minus 2 so 4 minus 2 is 2

so right now we have negative 1 over 4

cosine X sine cube X plus 3 over 4 and

earlier in this video we said that the

integral of sine squared was 1/2 X minus

1/4 sine 2x and of course plus C so

let's distribute the 3/4 everything

inside here so it's a negative 1/4

cosine X sine cube and then 3 over 4

times 1 over 2 that's gonna be 3 over 8

and then three fourths times 1/4 that's

3 over 16 sine to ax plus C so this is

the answer if you want to leave it in

that form

now I'm gonna adjust the answer I'm

gonna simplify it using the double angle

formula for sine so recall that we said

that sine X cosine X is 1/2 sine 2x

so I'm gonna rewrite sine cube as sine

ax times sine squared and everything

else I'm just gonna keep it the same for

now

now let's replace sine cosine with 1/2

sine X so I have negative 1/4 times 1/2

sine 2 X and then I'm going to replace

sine squared with something

I'm going to use the double angle

formula for cosine or you could say the

power reducing formula for sine squared

so you need to know that sine squared is

equal to 1/2 and then it's 1 minus

cosine 2x so that's the power reducing

formula for sine squared

and then everything else would be the

same so now I have a 4 times the 2 times

the 2 4 times 2 is 8 times 2 is 16 so

this is negative 1 over 16 and then sine

2x times 1 minus cosine 2x and then

everything else now I need to distribute

negative 1 over 16 sine 2x - 1 - cosine

2x so when we multiply it by 1 they're

just gonna be negative 1 over 16 sine 2x

and then if we multiply it by negative

cosine X is going to be positive 1 over

16 sine 2x cosine 2x

now let's combine like terms negative 1

over 16 minus 3 over 16 that's gonna be

negative 4 over 16 sine of 2x now early

we said that sine X cosine X is 1/2 sine

2x so therefore sine 2x cosine 2x must

be 1/2 sine 4x so if we double X to 2x

then 2x has to be double to 4x so let's

replace sine 2x cosine 2x with 1/2 of

sine 4x

now let's put this in order so we have 3

over 8 X and then for over 16 we can

reduce that to negative 1 over 4 sine 2x

and then 1 over 16 times 1 over 2 that's

1 over 32 sine 4x and then plus C so

this is another way in which you could

represent the answer so this is the

integral of sine to the 4 facts