hey,
I'm going through a chapter about integration but haven't really ever been taught it
besides the basic raise the power by one and divide by the new power...can someone please help me with the
rules of integration? please 
To make it easier for yourself, you should remember that integration is the opposite of deriving a function;
1.
 dx = -cos(x) + c)
This says that the anti-derivative of sin(x) is -cos(x). This is just the universal rule for it, and you can prove it by differentiating -cos(x), so the derivative of -cos(x) = -(-sin(x)) = sin(x).
2.
 dx = sin(x) + c)
The anti-diff of cos(x) is sin(x). You can prove it by deriving sin(x), which yields cos(x), so if you 'undo' (anti diff) cos(x) you should get back to the starting function, which was sin(x).
3.
^n dx = \frac{(ax+b)^{n+1}}{a(n+1)} + c)
, n=/= -1
This is just a rule that you need to memorise. Whenever you have a linear function INSIDE the brackets, like x+3 or 4-4x, and the power of the whole brackets is any number besides -1, you can use the above rule to integrate it.
Example:
^4 dx} = \int 2(3x-1)^{-4}dx)
Now, a = 3, b = -1, n = -4, plug these into the rule to get:
^{-4}dx = \frac{2(3x-1)^{-3}}{3(-3)} + c)
^3} + c)
4.
}dx = \frac{1}{a}log_{e}|ax+b| + c )
OR
}{f(x)} dx = log_{e}|f(x)| + c)
This one is like number 3, where the LINEAR expression inside the brackets is raised to the power of -1 when it is reciprocated. So when you have a non-fraction linear expression raised to the power of -1, you must use the above rule to integrate it.
Example:
}dx = 3\int \frac{1}{(x-1)} dx)
5.
This one is probably the most simple one, whenever you have e raised to the power of a number, just divide the euler expression by the number and add c to obtain the integral. You can once again prove that this is correct by differentiating the anti-diff:
Example:
 = \frac{e^{ax}}{a})
 = \frac{a}{a}e^{ax} = e^{ax})
So clearly the derivative of
equals e^{ax}, so this means that the anti-diff rule above holds.
I would recommend you to just keep practicing anti-differentiating some expressions, until you can fully memorise and utilise the rules above. Just to make things easier for you, try to put a poster of the rules on your wall for a week or two, this is what I did and it immensely helped me. Also for some additional help, just keep in mind when you anti-diff an expression, if you differentiate the result, it should lead back to the anti-diff original expression. Hope this helped xD
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