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Mathematics LibreTexts

4.2: Logs and Integrals

Recall that

\[ \int \dfrac{1}{x} dx = \ln |x| + C.\]

Note that we have the absolute value sign since for negative values of that graph of \(\frac{1}{x}\) is still continuous.

Example 1

Evaluate the integral

\[ \int \dfrac{dx}{1-3x}.\]

Solution

Let \(u = 1-3x\) and \(du = -3\, dx\).

The integral becomes

\[\begin{align}  -\dfrac{1}{3} \int \dfrac{du}{u} &=  \dfrac{1}{3}\ln |u| +C \\ &= -\dfrac{1}{3} \ln |1-3x| +C. \end{align}\]

Exercises

Evaluate the integrals of the following:

1)  \(\dfrac{1}{(x-1)}\)

2)  \(\dfrac{1}{(1-x)}\)

3)  \(\cot x\)

4)  \(\dfrac{(2x - 1)}{(x + 2)}\)          

5)  \(\dfrac{3x}{(x^2 + 1)^2}\)                     

6)  \(\dfrac{1}{x \ln x}\)

7)  \(\dfrac{1}{\sqrt{x - 1}}\)

8)  \(\dfrac{(x^2 + 2x + 4)}{(3x)}\)

9)  \(\dfrac{(x + 1)}{(x^2 + 2x)^3}\)

10)  \((4 - x)^5 \)

11) \(\dfrac{1}{\sqrt{3x}}\)

12) \(\tan x\)

13) \((\tan x)(\ln(\cos x))\)

14)  \(\sec x\)  (hint:  multiply top and bottom by \(\sec x + \tan x)\)

15)  \(\csc x\)  (hint: Use the formula \(\csc x = \sec (\pi/2 - x)\).    

Contributors

  • Integrated by Justin Marshall.