Integrals of Transcendental Functions: Natural Logarithms

 $\int\left(\frac{1}{x^3}\right)dx$  (Hint: This is a trick question. Be careful!)

• ln| $x^3$ | + C

•  $\frac{1}{x^2}$   + C

•  $-\frac{3}{x^4}$   + C

•  $-\frac{1}{2x^2}+C$  + C

 $\int\frac{1}{x}dx$  

• ln|x| + C

•  $-\frac{1}{x^2}$  + C

•  $\frac{1}{x^2}$  + C

• 1 + C

• $\frac{1}{\left(2x^2+6\right)^2}+C$

• $\ln\left|\frac{2x^2+6}{4x}\right|+C$

• $\frac{1}{2x^2+6}+C$

• $\ln\left|2x^2+6\right|+C$

 $\int\frac{2x}{x^2+1}dx$ 

•  $\ln\left|2x\right|$ + C

•  $\ln\left|x^2+1\right|$ + C

•  $\frac{1}{\left(x^2+1\right)^2}$ + C

•  $\frac{1}{2x}$  + C

Evaluate the indefinite integral.

• A

• B

• C

• D

Which of these can be integrated using u-substitution?

• A

• B

• both

• neither

Hint: Separate into two integrals first!

• $-3\ln\left|\cos x\right|+2\cos x+C$

• $-3\ln\left|\sin x\right|-2\sin x+C$

• $-3\ln\left|\cos x\right|-2\cos x+C$

• $3\ln\left|\cos x\right|-2\sin x+C$

 $\int\frac{1}{2x}dx=$  

•  $\ln\left|2x\right|+c$  

•  $\frac{1}{2}\ln\left|2x\right|+c$  

•  $2\ln\left|2x\right|+c$  

•  $2\ln\left|x\right|+c$  

 $\int\frac{x^2}{x^3+4}dx=$  

•  $\frac{1}{3}\ln\left|x^3+4\right|+c$  

•  $\ln\left|x^3+4\right|+c$  

•  $3\ln\left|x^3+4\right|+c$  

•  $x^2\ln\left|x^3+4\right|+c$  

 $\int\frac{1}{2-x}dx=$  

•  $\ln\left|2-x\right|+c$  

•  $\frac{1}{2}\ln\left|2-x\right|+c$  

•  $2\ln\left|2-x\right|+c$  

•  $-\ln\left|2-x\right|+c$