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. You may want to use the table of integrals and the properties of integrals in this site. In what follows, C is a constant of integration and can take any constant value.
Example 1Evaluate the integral
Solution to Example 1:
Let U = ln(x) and V' = 1 and use integration by parts. Hence U ' = 1/x and V = x
Example 2Evaluate the integral
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 (found in example 1) function to obtain
∫ ln(2x + 1) dx = (1 / 2) ∫ ln u du = (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 3Evaluate the integral
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 4Evaluate the integral
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.
ExercisesEvaluate 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