Logs and Integrals

Logs and Integrals

Recall that

       

Note that we have the absolute value sign since for negative values of that graph of 1/x is still continuous.

Example



Evaluate the integral

       

Solution

 let u = 1-3x, du = -3dx

The integral becomes

       

Exercises



Evaluate the integrals of the following:  (for answers, place your mouse over the yellow rectangle)

A)  1/(x - 1)                        B)  1/(1-x)                            C)  cot x

ln | x - 1 | + C                    - ln | x - 1 | + C                    ln | sin x | + C

D)  (2x - 1)/(x + 2)             E)  3x/(x2 + 1)2                      F)  1/(x ln x)

2x - 5ln | x + 2 | + C                    -3 / (2 (x^2 + 1) ) + C                        ln | ln (x) | + C

G)  1/sqrt(x - 1)                  H)  (x2 + 2x + 4)/(3x)            I)  (x + 1)/(x2 + 2x)3  

2 sqrt ( x - 1 )                    -1 / 3 ( x^2 / 2 + 2x + 4 ln | x | ) + C                        1/4 ( x^2 + 2x )^(-2) + C

J)  (4 - x)5                           K) 1/sqrt(3x)                         L)  tan x

-1/6 (4 - x)^6                    2sqrt(x) / sqrt(3)                         ln | sec x | + C

M) (tan x)(ln(cos x))                    N)  sec x  (hint:  multiply top and bottom by sec x + tan x)

-[ ln (sin x) ] ^ 2 / 2 + C                                ln | sec x + tan x | + C

O)  csc x  (hint: Use the formula csc x = sec(p/2 - x)      

- ln | csc x + cot x |

 

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