Integrating Sin2xCosx

In the Solomon paper L for C4 (OCR), theres the following question:

Evaluate
(with limits b = pi/3, and a = 0)

∫ Sin2x.Cosx dx

In the markscheme they've solved it as:

∫2Sinx.Cos²x dx

Which i assume they got from expanding Sin2x = 2SinxCosx

And they then jump to:

[(-2/3).Cos³x] with the limits from above.

I assume what they've done here is use a substitution of u=Cosx, so du/dx = -Sinx, so the integral becomes:

∫(2Sinx.u²) / -sinx = ∫ -2u² du = [ (-2/3)u³ ] = [ (-2/3)Cos³x ]

However i confused myself here thinking it wasnt correct to integrate Cos²x like this, as i thought that when integrating cos²x and sin²x, i had to use the double angle identities.

So is what i've done here the correct way to solve it, even though i didnt think it was correct maths to integrate cos²x this way? And whats the best way for me to identify which integration method to use on this sort of question? (I thought i had to integrate this by parts when i first saw it).

Reply 1

dvs

You weren't integrating cos²x here -- it was cos²x.sinx, which is very different.

The markscheme used the "function and its derivative" rule, which is essentially, as you spotted, using the substitution u=cosx with du/dx=-sinx.

If you did have cos²x by itself, then you would've needed to use the appropriate double-angle identity for cos2x.

Reply 2

naystar

Indeed, spotting a function and derivative is handy for integrating. And a substitution in this case is fine. All you've got to practise is being able to spot such cases!

Reply 3

Migraine

Ok, thanks for the help.

Reply 4

SsEe

Alternatively,
Using sin(A)cos(B) = 0.5[ sin(A+B) + sin(A-B) ]
Sin(2x)Cos(x) = 0.5[ sin(3x) + sin(x) ]
Which integrates to
0.5[ -(1/3)cos(3x) - cos(x) ] + c
Limits:
0.5[ (1/3) - (1/2) + (1/3) + 1 ]
= 7/12

Reply 5

Font

Migraine

∫2Sinx.Cos²x dx
Which i assume they got from expanding Sin2x = 2SinxCosx
And they then jump to:
[(-2/3).Cos³x] with the limits from above.

its done my a method called "recognition" - you merely look at the integral and work backwards saying "what function must i differentiate wto get the integral"

wol ah,

Maths made easy :smile:

regarding method - what tehy tell you to use - if its an easy one, they'll explect you dto do it..ones like that they may well give you a substitution because the majority of people cant do recognition

pk

Reply 6

seankhn

Is that the same as "standard patterns"?