Help I don’t understand integration

Can someone explain how to do these I don’t get it

What are the answers supposed to be?

The answers are all correct on there but I don’t the working

That was why I asked because I couldn't see a mistake.

Which bit don't you get?

I don’t get how to do the first q…
how do I get 11…
do I plus one to the f(x)
I’m so confused

How do I know it’s 3+ (9-1)
?

It's because integral(f(x) + 1) = integral(f(x)) + integral (1)
The value of integral(f(x)) is given to you, so you just need to add integral (1)

Ahh I see thank you!!
Also what about the second one… I’m so sorry for being annoying but I’m really dumb
I don’t get why they put the minus there isn’t it just times by 2 because it’s stretch in y direction

If you look closely you'll see that the limits of integration have been reversed. That has the effect of negating the original integral.

I don't get for the first one how integrating 1 just gives you 9 - 1. When you integrate 1 you get x, so surely it'd be 9x - x, so 8x not 8? I got the second one, but can someone explain this with the first one?

I think its just a bit of sloppy writing / missing a + as you get
Int_1^9 f(x) dx + Int_1^9 1 dx
the first term is 3 as your given it, the second definite integral is 9-1 as you get x before subbing in the limits.

That's my bad. I genuinely am so stupid, I forgot the 9 and 1 just went in place of x and for some reason thought it'd be 9x - x as if it were expanding the brackets. Thank you for explaining that to me.

With questions like this it's almost worth taking a step back and asking "what does the integral represent?" If you integrate '1' between limits of 1 and 9 then you're basically calculating the area of a rectangle with height 1 and width 8 ( = 9 - 1), so the extra bit of area added to the original integral is just going to be 8.