Integration (indifinite integrals) Week 2 18 & 19/12/2024

 $Find\ the\ integral\ \int\left(x+\frac{3}{x^2}\right)^2dx$  

• a)  $\frac{x^3}{3}+\ln x-\frac{3}{x^3}+c$  
  

• b)  $\frac{x^3}{3}+6\ln x-\frac{3}{x^3}+c$  
  

• c)  $\frac{x^{-3}}{3}+6\ln x-\frac{3}{x^2}+c$  
  

• d)  $\frac{x^2}{3}+\ln x-\frac{4}{x^4}+c$  
  

Using integral by parts, solve

 $\int xe^{-x}dx$  

• a)  $e^{-x}\left(-x-1\right)+c$  
  

• b)  $xe^x\left(2x-1\right)+c$  
  

• c)  $e^{2x}\left(\frac{3}{2}x+1\right)+c$  
  

• d)  $3e^x\left(2x+3\right)^2+c$  
  

 $Solve\ \ \ \int\left(\frac{\left(x^2-1\right)}{x^2\left(2x+1\right)}\right)dx$ 

(Use integral by parts)

• a)  $\frac{3}{2}\ln\left(\frac{2x}{2x+1}\right)+c$  
  

• b)  $2\ln3x+x^2-2\ln\left|3x-5\right|+c$  
  

• c)  $\ln2x-\frac{1}{x^2}-3\ln\left|2x+1\right|+c$  
  

• d)  $2\ln x+\frac{1}{x}-\frac{3}{2}\ln\left|2x+1\right|+c$  
  

Use the substitution   $u=x^2$  , find

 $\int\left(\frac{3x}{x^4-4}\right)dx$  

• a)  $3\ln\left|x^2-2\right|+5\ln\left|x^2+2\right|+c$  
  

• b)  $\ln\left|x^2-2\right|-5\ln\left|x^2+2\right|+c$  
  

• c)  $\frac{3}{8}\ln\left(\frac{x^2-2}{x^2+2}\right)+c$  
  

• d)   $\frac{2}{3}\ln\left|\frac{x^2+2}{x^2-2}\right|+c$  
  

• 1.27

• 1.52

• 1.36

• 1.82

• 2.41

• 2.39

• 2.07

• 2.27