Integration, please help!

Question

Evaluate the integral  .

Options:

1. 
2. 
3. x - 1 + ln|x| + C
4. 

Answer 1

Hi Selena, this integral is a little tricky but here goes:

Make the substitution u = x + 1 du = dx We'll also need that x = u - 1

∫(u-1)²/u du now expand the numerator

∫(u² - 2u + 1)/u du now break up into 3 terms

∫(u²/u -2u/u + 1/u)du

∫(u -2 + 1/u)du = 1/2u² - 2u + ln|u| + C now sub back to x

= 1/2(x + 1)² - 2(x + 1) + ln|x + 1| + C now expand and combine like terms

= 1/2(x² + 2x + 1) -2x -2 + ln|x+1| + C

= 1/2x² + x + 1/2 -2x -2 + ln|x+1| + C

= 1/2x² - x - 3/2 + ln|x+1| + C

which actually look like one of the choices! What a relief.

The real trick here that is kind of different is solving u=x+1 for x so we can substitute the x² term. Integration is a big bag of tricks and you have to work a lot of them to know what to do. Good Luck!