My grandson asked me this one - but i need help! Integrate sin5x cos2x using a method other than integration by parts. Any ideas?

Question

My grandson asked me this one - but i need help! Integrate sin5x cos2x using a method other than integration by  parts. Any ideas?

amistre64 · Answer

use the trig identitys of cos addition and cosine subtraction

amistre64 · Answer

what is cos(a+b)  and cos(a-b)?

amistre64 · Answer

bah .. sin cos is in sin addition and subtraction lol

welshfella · Answer

oh ok 
yeas i know what they are

amistre64 · Answer

what is ... sin(a+b) and sin(a-b)  those should work out better

welshfella · Answer

so use sin (a +b) and sin (a -b) ?

amistre64 · Answer

type them for me and lets see how they are useful

amistre64 · Answer

by the way, this is the impetus behind fourier series ...

welshfella · Answer

sin (a + b) = sina cosb + cosa sinb
sin(a -b) = sina cosb - cosa sinb

welshfella · Answer

/oh !  add them together?

amistre64 · Answer

yes ;)

now when we add the 2 togheter,  the rightest terms go to zero 

sin(a+b) + sin(a-b) = 2 sina cosb

welshfella · Answer

right i got it now
plug in 5x and 2x for a and b

amistre64 · Answer

very good

welshfella · Answer

and we'll get the sum of sin and cos

welshfella · Answer

clever stuff

amistre64 · Answer

$$sin(a)cos(b)=\frac12[sin(a+b)+sin(a-b)]$$
$$sin(5x)cos(2x)=\frac12[sin(7x)+sin(3x)]$$

welshfella · Answer

yea

welshfella · Answer

now its easy to integrate

thanx

amistre64 · Answer

youre welcome :)

welshfella · Answer

I'm not sure if i did this in the past. Must have i guess

amistre64 · Answer

it wasnt covered that well in my courses, i only really learned it when i was going over fourier series.  they use this techique to find the coefficients of their series.

welshfella · Answer

I did fourier series but its now so hazy  lol!

amistre64 · Answer

heheh, oh i know that feeling all to well

welshfella · Answer

yea  ty