Evaluate Integrals Involving Logarithms - Tutorial

Evaluate integrals involving natural logarithmic functions: A tutorial, with examples and detailed solutions. Also exercises with answers are presented at the end of the tutorial.

In what follows, c is a constant of integration and can take any constant value. You may want to use the table of integrals and the properties of integrals in this site.



Example 1: Evaluate the integral

ln(2x + 1) dx


Solution to Example 1:

Substitution: Let u = 2x + 1 which leads to du / dx = 2

or du = 2 dx or dx = du / 2, the above integral becomes


ln(2x + 1) dx = (1/2) ln u du

We now use integral formulas for ln x function to obtain

ln(2x + 1) dx = (1 / 2) [u ln u - u] + c

We now substitute u by 2x + 1 into the above to obtain

ln(2x + 1) dx = (1 / 2)(2x + 1) ln (2x + 1) - (1 / 2)(2x + 1) + C

= (1 / 2)(2x + 1) ln (2x + 1) - x - 1/2 + C

= (1 / 2)(2x + 1) ln (2x + 1) - x + k , where k = c - 1/2 and is a constant.

Check: Differentiate (1 / 2)(2x + 1) ln (2x + 1) - x + k and see that you obtain ln(2x + 1) which is the integrand in the given integral. This is a way to check the answer to indefinite integrals evaluation.



Example 2: Evaluate the integral

x ln x dx


Solution to Example 2:

Let f(x) = ln x and g ' (x) = x

which gives f'(x) = 1 / x and g(x) = x 2 / 2.

Using the integration by parts

f(x) g '(x) dx = f(x) g(x) - f '(x) g(x) dx , we obtain


x ln x dx = [ x 2 / 2] ln x - [ x 2 / 2 ] [1 / x] dx

= [ x
2 / 2] ln x - [ x / 2 ] dx =

= [ x
2 / 2] ln x - x 2 / 4 + c.

Practice: Differentiate [ x
2 / 2] ln x - x 2 / 4 + c to obtain the integrand x ln x in the given integral.



Example 3: Evaluate the integral

ln(x) / x dx


Solution to Example 3:

Let u = ln x so that du / dx = 1 / x; the given integral can be written as

=
u du

Integrate to obtain

= u
2 / 2 + c

Substitute u by ln x

= [ln x]
2 / 2 + c

As an exercise, check the final answer by differentiation.

Exercises: Evaluate the following integral.

1. x 3 ln x dx

2. [x - ln x] dx



Answers to Above Exercises

1. x 4 ln x / 4 - x 4 / 16 + c

2. -x ln x + x 2 / 2 + x + c

More references on integrals and their applications in calculus.