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Need help with why we do this.

Howdy

I am struggling with this example and the general method for this type of integration.

I can't explain why we 'consider y = sin(2x + 3)'

Something about it makes it easier as then you check if this is correct by differentiating it.

What I'd like help with is a better explanation of this please as I don't feel confident in what I am doing/saying to myself.

WhatsApp Image 2024-12-30 at 16.20.38.jpeg

WhatsApp Image 2024-12-30 at 16.20.38.jpeg

Reply 1

mqb2766

Original post

by makin

Howdy

I am struggling with this example and the general method for this type of integration.
I can't explain why we 'consider y = sin(2x + 3)'
Something about it makes it easier as then you check if this is correct by differentiating it.
What I'd like help with is a better explanation of this please as I don't feel confident in what I am doing/saying to myself. WhatsApp Image 2024-12-30 at 16.20.38.jpeg

Its essentially doing the reverse chain rule where you "know" that the integral of cos is sin, but the 2x+3 is the difficult bit. So guess that its sin(2x+3) and differentiate ("forwards" chain rule) to see how that compares with the given function. In this case youre off by a factor of 2 so ...

Reply 2

makin

Original post

by mqb2766

Thanks. I've made notes on it and I can answer questions from the book. I just don't feel I can explain why we jump to 'consider y = sin(2x + 3)'

I've put that integrating cos x gives sin x, so try sin (2x + 3)

Would you define this as something like...

'we consider the integration of the outer part to help make this be done easier. As we can then see, via differentiation, whether this takes us correctly back to our original question'

Screenshot 2024-12-30 at 17.14.11.png

WhatsApp Image 2024-12-30 at 16.20.38.jpeg38.4KB

Reply 3

mqb2766

Original post

by makin

Thanks. I've made notes on it and I can answer questions from the book. I just don't feel I can explain why we jump to 'consider y = sin(2x + 3)'
I've put that integrating cos x gives sin x, so try sin (2x + 3)
Would you define this as something like...
'we consider the integration of the outer part to help make this be done easier. As we can then see, via differentiation, whether this takes us correctly back to our original question' Screenshot 2024-12-30 at 17.14.11.png

I think that looks ok and the formula you quote at the bottom is just the chain rule where the inner function (argument) is linear (as they note with the "watch out"), hence the a or 1/a multipliers on the derivative or integral. Really the reverse chain rule is used in such cases where you can guess what integral is so cos->sin, then a simple scaling by 1/a to account for the linear inner function/argument. More generally, you do this type of integration problem using substitution but for simple cases like this, its the same as the reverse chain rule.