# 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:

int(du)/u=ln\ |u|+K

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:

int(f^'(x))/(f(x))dx=ln\ |f(x)|+K

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

int(2x^3)/(x^4+1)dx

### Example 2

int_0^(pi//4)(sec^2x)/(4+tan x)dx

Here is the curve y=(sec^2x)/(4+tan x):

### Example 3

int(dx)/(x\ ln\ x

### Example 4

The equation

t=int(dv)/(20-v)

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.

## Exercises

Integrate each of the given functions:

### Exercise 1

int(dx)/(x(1+2\ ln\ x)

### Exercise 2

int_1^2(x^2+1)/(x^3+3x)dx

Here's the graph of y=(x^2+1)/(x^3+3x):

### Exercise 3

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:

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