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. We take the opposite of the derivative of the logarithmic function to solve such cases:

math expression

The math expression (absolute value) signs around the u are necessary since the log of a negative number is not defined.

We can also write the formula as:

math expression

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.

Example 1: math expression


Answer

Example 2: math expression


Answer

Example 3: math expression


Answer

 

Example 4:

The equation

math expression

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.


Answer

 

Exercises

Integrate each of the given functions:

1. math expression


Answer

2. math expression



First, the LiveMath solution to this problem.

LIVEMath


Answer

 

3. The electric power p developed in a certain resistor is given by

math expression

where t is the time. Express p as a function of t.


Answer



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