How to know the difference between... (1 Viewer)

[1722]

okay so this is probably the stupidest question ever but:

How do you know if the derivative you are integrating will be a log/exponential function or a err simple function (if the derivative is in the form of a fraction)?
For example if it tells you to integrate:

x²/x³+7

or

x²-2x²+3/3x⁵

How do you know which one will integrate into a log/exponential or just a regular function?

 

[SharkeyBoy]

look at the powers. if the denominator has a higher power of one more than the numerator, then it integrates to a log.
is that what you're asking?

 

[1722]

Yes omg! Thank you! It's been bothering me for ages and I couldn't find where the difference was! Thank you sooo much!
P.S Nobody taught me when I was learning it in school

 

[SharkeyBoy]

hahaha nws

 

[bedpotato]

It's not always the powers.

When you differentiate ln(f(x)), the derivative is f'(x)/f(x)
So if you integrate f'(x)/f(x), wouldn't it be ln(f(x)) since integration is the reverse of differentiation.

The same rule applies when inegrating functions such as x^2/x^3 + 7, where f(x) is x^3 + 7 and f'(x) is x^2

E.g. If the question asks you to integrate cosx/sinx, you can't look at the powers.

Instead, differentiate the denominator, d/dx(sinx) = cosx
If the derivative of the numerator is the same as the numerator, use log.