INTEGRATION: questions (1 Viewer)

[Rahul]

1. when integrating by parts, which one do you choose to integrate or leave/differentiate. is there a set thing to this or any type of function to look out for either option?

2. integrate: (cos3x)(sinx).dx

2:
#(cos3x)(sinx).dx
= 1/2 #(sin2x+sin4x).dx
= 1/2 (-1/2.cos2x - 1/4.cos4x)
= -1/4.cos2x - 1/8.cos4x

the answer in the book says, 1/4.cos2x - 1/8.cos4x
???

i used the, cos(a)sin(b) = sin(a+b) - cos(a-b) indentity

edit: # = integral sign

 

[Affinity]

by parts generally(there are exceptions):
you want to differentiate polynomial parts, inverse trig parts and logarithm parts.

integrate trig and exponentials. hmm basically it..


2)the book's right... you're wrong. revise your trigonometry identities

 

[Rahul]

#(cos3x)(sinx).dx
= 1/2 #(sin2x+sin4x).dx
= 1/2 (-1/2.cos2x - 1/4.cos4x)
= -1/4.cos2x - 1/8.cos4x

can you show me where the mistake is?
thanks

edit: sorry late night, i see it i think.

 

[Affinity]

I will expand on the by parts bit:
basically... you don't want to get something more complicated than you started with. and the following indicates complexity

high powers are more complicated than low powers

irrational terms (squareroots cube roots etc) are more complicated than terms with integer indices

transcedental functions (trig, exponential, log, inverse trig) are more annoying than polynomials


and usually it's the integration which produces the mess, but integrating exponential will give exponential, integrating trigs will give trigs(sine and cosine I mean) without raising the power, so integrating these are good.

differentiating polynomials is good because it reduces power, differentiating log, inverse trig is good because it changes it into something simpler.

 

[mercury]

1. When you are integrating, if you do it using the let x = u y = dv method, you want to choose u and dv such that uv is not equal to a constant. You can derive that from definition of IBP

eg. if you integrate xlnx, it wouldn't work if you let u = x and dv = lnx

2. sinU - sinV = 2cos(U+V/2)sin(U-V/2)
U = 4x V=2x

cos3x * sinx = 1/2(sin4x - sin2x)
= -1/8cos4x + 1/4 cos2x

this should be right answer.

 

[blah]

Yeh the books right...

I think you got the trig identities wrong.

2sin(x)cos(3x) = sin(-2x) + sin(4x) so
sin(x)cos(3x) = -1/2sin(2x) + 1/2sin(4x) and not 1/2sin(4x) + 1/2sin(2x)

 

[loser]

Originally posted by Rahul
i used the, cos(a)sin(b) = sin(a+b) - cos(a-b) indentity

huh?

 

[Rahul]

thanks affinity

blah; 2sin(x)cos(3x) = sin(-2x) + sin(4x)

why is it sin(-2x)?

Originally posted by loser
huh?

i typed it up wrong, cos(a)sin(b) = sin(a+b) - sin(a-b). products to sums/differences.

edit: i know exactly what i did wrong.

 

[Rahul]

thanks.

hey it was a lil bit tricky

 

[deyveed]

Originally posted by Rahul

i used the, cos(a)sin(b) = sin(a+b) - cos(a-b) indentity

That trig identity is not in the syllabus right?
If it isn't, would there always be another way to answering the questions without using the identity?
Do you lose marks if you do use the identity?

 

[Rahul]

i dont think so, it was directly in the arnold book.

 

[deyveed]

In the Excel Maths Ext1 formula summary section it says its not in the syllabus

 

[Rahul]

eheh this is ext2

 

[ricenoodles]

is this identity in the syllabus:

cosa + cosb = 2cos[(a+b)/2]cos[(a-b)/2]

 

[blah]

2sin(a)cos(b) = sin(a-b) + sin(a+b)

 

[Constip8edSkunk]

its not in the syllabus but then riemanns hypothesis isnt aither

actually bad comparison... the trig identities are quite useful in general 4U though...

 

[underthesun]

By seeing the pattern to the force and compound angles, I think you can remember those formulas easily..