2. Integration: The Basic Logarithmic Form
by M. Bourne
The general power formula that we saw in Section 1 is valid for all values of n except n = −1.
If n = −1, we need to take the opposite of the derivative of the logarithmic function to solve such cases:
The `|\ |` (absolute value) signs around the u are necessary since the log of a negative number is not defined. If you need a reminder, see absolute value.
We can also write the formula as:
In words, this means that if we have the derivative of a function in the numerator (top) of a fraction, and the function in the denominator (bottom) of the fraction, then the integral of the fraction will be the natural logarithm of the function.
Here is the curve `y=(sec^2x)/(4+tan x)`:
The shaded region represents the integral we just found.
`int(dx)/(x\ ln\ x`
comes from considering a force proportional to the velocity of an object moving down an inclined plane. Find the velocity, v, as a function of time, t, if the object starts from rest.
This is the graph of the velocity of the sliding object at time t.
Integrate each of the given functions:
`int(dx)/(x(1+2\ ln\ x)`
Here's the graph of `y=(x^2+1)/(x^3+3x)`:
The shaded region is the integral we just found.
The electric power p developed in a certain resistor is given by
`p=3int(sin\ pi t)/(2+cos\ pi t)dt`
where t is the time. Express p as a function of t.
Here's the graph of our solution:
The graph of the power p at time t (using K = 2).