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C4 Integration question

How do you integrate sin2xcosx, with respect to x?

Thanks in advance :smile:
Reply 1
Avatar for Zacken
Zacken
22
Original post by jasminetwine
How do you integrate sin2xcosx, with respect to x?

Thanks in advance :smile:


2sin⁡xcos⁡2x2 \sin x \cos^2 x, now use the sub u=cos⁡xu = \cos x.

Reasoning behind the sub: we can see that the integral is of the form derivative * function, that always suggests a sub u = function since the derivatives then cancel.
(edited 8 years ago)
Reply 2
Avatar for B_9710
B_9710
18
You may want to consider the derivative of (cosx)^3 to find integral of 2sinx(cosx)^2.
Original post by Zacken
2sin⁡xcos⁡2x2 \sin x \cos^2 x, now use the sub u=cos⁡xu = \cos x.

Reasoning behind the sub: we can see that the integral is of the form derivative * function, that always suggests a sub u = function since the derivatives then cancel.


In case you're wondering how we got to 2sin⁡xcos⁡2x2 \sin x \cos^2 x, we don't like the sin(2x) bit - we want it in terms of sin(x). Is that correct?
Reply 4
Avatar for Zacken
Zacken
22
Original post by spotify95
In case you're wondering how we got to 2sin⁡xcos⁡2x2 \sin x \cos^2 x, we don't like the sin(2x) bit - we want it in terms of sin(x). Is that correct?


Yep. I didn't bother mentioning that because I helped the user out on another question today and she seemed comfortable with converting sin 2x to 2 sin x cos x and vice versa. :yep:
Original post by Zacken
Yep. I didn't bother mentioning that because I helped the user out on another question today and she seemed comfortable with converting sin 2x to 2 sin x cos x and vice versa. :yep:


Thought so - I remember the 2sin(x)cos(x) thing well from my days of C4.
Original post by spotify95
In case you're wondering how we got to 2sin⁡xcos⁡2x2 \sin x \cos^2 x, we don't like the sin(2x) bit - we want it in terms of sin(x). Is that correct?


Why don't we like sin2x? I don't understand :smile: we could differentiate it to 2cos2x couldn't we?
(edited 8 years ago)
Original post by jasminetwine
Why don't we like sin2x? I don't understand :smile: we could differentiate it to cos2x/2 couldn't we?


It would go to -1/2 cos(2x)

In this case, it's just easier to deal with in terms of sin(x) and cos(x) because there are already terms in cos(x) anyway :yep:

2cos(2x) would be what would happen if you differentiated sin(2x).
(edited 8 years ago)
Reply 8
Avatar for Zacken
Zacken
22
Original post by jasminetwine
Why don't we like sin2x? I don't understand :smile: we could differentiate it to 2cos2x couldn't we?


When you're integrating product of trig functions, you almost always want them to have the same argument, i.e: all in terms of trig(2x) or trig(x) or trig(3x) or whatever and in this case it's easier to convert sin 2x into trig(x) instead of cos(x) into trig(2x)
Original post by Zacken
When you're integrating product of trig functions, you almost always want them to have the same argument, i.e: all in terms of trig(2x) or trig(x) or trig(3x) or whatever and in this case it's easier to convert sin 2x into trig(x) instead of cos(x) into trig(2x)


Ahh I see!

This is what I got (again, please excuse my messy workings!) :smile:

image.jpg
Reply 10
Avatar for Zacken
Zacken
22
Original post by jasminetwine
Ahh I see!

This is what I got (again, please excuse my messy workings!) :smile:

image.jpg


Perfect. :smile:
Thank you so much for your help and patience both of you!

I really can't tell you how grateful I am!
Reply 12
Avatar for Zacken
Zacken
22
Original post by jasminetwine
Thank you so much for your help and patience both of you!

I really can't tell you how grateful I am!


Don't sweat it. :-)

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