Help with a level integration question-by parts Watch

alex.scholes28
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Attachment 823494
Got a bit stuck on this integration question, I think they’ve used by parts-which I’ve tried to do, and let u= ln x and dv/dx=k. I don’t understand why they have got the integral of k dx (next to second red tick) rather than integral of k •1/x dx (as du/dx would be 1/x)?
Hope this explanation made sense☺️
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Sinnoh
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Picture quality is too small - I can't tell what the question is.
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alex.scholes28
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(Original post by Sinnoh)
Picture quality is too small - I can't tell what the question is.
The rate of change of y with respect to x is proportional to ln x. Fine y when x=8

Can you see the working out/markscheme clear enough?
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Sinnoh
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(Original post by alex.scholes28)
The rate of change of y with respect to x is proportional to ln x. Fine y when x=8

Can you see the working out/markscheme clear enough?
When you do IBP, and you integrate one of the parts, you keep that integrated part for the second integral. So you've got  \int \frac{kx}{x}, because you integrate k to get kx. They've skipped out the intermediate step of cancelling out the x's.
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alex.scholes28
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(Original post by Sinnoh)
When you do IBP, and you integrate one of the parts, you keep that integrated part for the second integral. So you've got  \int \frac{kx}{x}, because you integrate k to get kx. They've skipped out the intermediate step of cancelling out the x's.
Aaah Thankyou so much! Think my brains been fried by revision
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Sinnoh
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(Original post by alex.scholes28)
Aaah Thankyou so much! Think my brains been fried by revision
Np. Take a break perhaps lol
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alex.scholes28
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(Original post by Sinnoh)
Np. Take a break perhaps lol
Am gonna do that now, need to have a rest or I’ll just keep making silly mistakes
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