C4 - edexcel INTEGRATION!

Are we expected to know how to integrate a^x? And the proof for it?

Probably.

Have a go at it, it's nothing over A-Level maths level and there is a hint in your formula booklet that you can use:

Oh nice so if i ever have a question on it i can just refer to this and then use integration via substitution to get:
(1/lna)a^x + c?

when your integrating:
e^u (1/lna) du. You're alowed to remove the 1/ln1 outside the integral right? is it only when you have a multiple you can remove it to the other side?

Yep.



1/ln(a) can be moved outside, yes. Because it is a constant in this process of integration, whereas $u$ is not.

Are we expected to know how to integrate a^x? And the proof for it?

what do you mean by constant? I thought only multiples can be removed eg. integrating 2/(ax+b) = remove 2 to outside the integral, work out 1/(ax+b), then multiply answer (1/a)ln(ax+b) by 2?

Multiple are constants, but we are not restricted to only multiples.

but what about something like: (e^x)(2^x) wouldnt 2^x classify as a constant as we are multiplying it to e^x?

No because it's being raised to a variable.

Being able to recognise what's a constant and what's a variable is extremely important in integration as it is the difference between making your life easy, or hell.

Okay so bassically any number being added/multiplied will count as a constant so it can be removed. But anything with a letter cannot?

It's not that basic so your generalisation is not quite correct. Also 'anything with a letter' doesn't mean that that will be a variable. Clearly, in the example you've done, $a$ is a letter but it's a constant.

In the process of integration, we take all the 'letters' other than the letter we are integrating with respect to as constants (unless they are functions of $x$, as then these are just expressions in terms of $x$ in 'disguise'. Whether they can be removed entirely from the integral is an entirely different matter to deal with. I.e. integrating $ax$ means that you can factor the $a$ out of the integral, but integrating something like $a^x$ means you cannot do that since the two aren't separable.

Oh I see so if it is seperable and will give the same answer (if not removed from the integration) then its considered a constant. So when we did ln(a) why are we allowed to remove it from the integral? wouldnt it need to be integrated as well?