16. Integration and differential equations

∫ f’(x)/f’’(x) dx =

ln|f(x)| + C

when using integration by substitution for a definate integral, what do you do if your lower limit becomes becomes greater than the upper limit?

keep the order of the limits the same even if the upper limit is smaller than the lower limit

how to choose what to use for u in integration by parts

LIATE for u - based on how easy it is to differentiate

L - logarithms

I - inverse trig

A - algebra

T - trig

E - exponential

what method of integration do you use for integrating lnx

intergration by parts using u as lnx and v as 1

How to integrate e^x (sinx)

Use integration by parts twice

then rearrange the equation such that 2∫e^x sinx =

what to do if you ever get stuck in a loop when using integration by parts

remeber that there are two sides of the equation, could you consider rearanging

How to complete a thorough proof of a point of inflection

state at point of inflection f’’(x) = 0

calculate second derivative

solve for value of x

if multiple values check against domain to ellimante values

to confirm point of inflection check either side on the x value with the second deivative to confirm concavity changes