Integration

[l>]

Hi, I have a quick question about integration. I don't really understand when to use substitution and when to use integration by parts. Is there some kind of rule for that? Or how do you know which to use?

[eross]

Hi, I have a quick question about integration. I don't really understand when to use substitution and when to use integration by parts. Is there some kind of rule for that? Or how do you know which to use?

 

I don't think there's a rule (sometimes both parts AND substitution work!) but it's useful to know some "tricks". Really, the only way is to practice so much that you start knowing which to use just by experience. I below I write a list I've compiled: 

 

integration by substitution techniques: 

if function contains sqare root of f(x), try substituting u=f(x)

if function contains ln x, try substituting u=ln x

if function contains square root(a² - x²), try substituting u=a*sin x

if function contains square root( x² - a² ), try substituting u=a*sec x

if function contains square root(x² + a²) or just (x² + a² ), try substituting u=a* tan x

 

to integrate cos²x, do NOT do parts (it works, but it's awfully long and takes time); use the identity cos (2x) = 2cos²x - 1 to substitue the cos^2 by a cos(2x). 

 

memorize the antiderivative of tan(x)-- it shows up at least once per exam. integral of tan(x) dx = -ln[ abs. value (cos x)] or ln[abs. value (sec x)] 

 

to integrate ln, do parts integrating (1) (ln(x)) 

 

hope these helped!

[ibstudent77]

 

Hi, I have a quick question about integration. I don't really understand when to use substitution and when to use integration by parts. Is there some kind of rule for that? Or how do you know which to use?

 

I don't think there's a rule (sometimes both parts AND substitution work!) but it's useful to know some "tricks". Really, the only way is to practice so much that you start knowing which to use just by experience. I below I write a list I've compiled: 

 

integration by substitution techniques: 

if function contains sqare root of f(x), try substituting u=f(x)

if function contains ln x, try substituting u=ln x

if function contains square root(a² - x²), try substituting u=a*sin x

if function contains square root( x² - a² ), try substituting u=a*sec x

if function contains square root(x² + a²) or just (x² + a² ), try substituting u=a* tan x

 

to integrate cos²x, do NOT do parts (it works, but it's awfully long and takes time); use the identity cos (2x) = 2cos²x - 1 to substitue the cos^2 by a cos(2x). 

 

memorize the antiderivative of tan(x)-- it shows up at least once per exam. integral of tan(x) dx = -ln[ abs. value (cos x)] or ln[abs. value (sec x)] 

 

to integrate ln, do parts integrating (1) (ln(x)) 

 

hope these helped!

 

Sorry this is unrelated, is this tested in SL Math as well?

[IBsurvivor98]

 

 

Hi, I have a quick question about integration. I don't really understand when to use substitution and when to use integration by parts. Is there some kind of rule for that? Or how do you know which to use?

 

I don't think there's a rule (sometimes both parts AND substitution work!) but it's useful to know some "tricks". Really, the only way is to practice so much that you start knowing which to use just by experience. I below I write a list I've compiled: 

 

integration by substitution techniques: 

if function contains sqare root of f(x), try substituting u=f(x)

if function contains ln x, try substituting u=ln x

if function contains square root(a² - x²), try substituting u=a*sin x

if function contains square root( x² - a² ), try substituting u=a*sec x

if function contains square root(x² + a²) or just (x² + a² ), try substituting u=a* tan x

 

to integrate cos²x, do NOT do parts (it works, but it's awfully long and takes time); use the identity cos (2x) = 2cos²x - 1 to substitue the cos^2 by a cos(2x). 

 

memorize the antiderivative of tan(x)-- it shows up at least once per exam. integral of tan(x) dx = -ln[ abs. value (cos x)] or ln[abs. value (sec x)] 

 

to integrate ln, do parts integrating (1) (ln(x)) 

 

hope these helped!

 

Sorry this is unrelated, is this tested in SL Math as well?

 

[eross]

Sorry this is unrelated, is this tested in SL Math as well?

 

 

 

 

Um, I know that you guys definitely do integrals, and you do some substitutions (I'm not sure how many of the ones above apply to SL).  Have you done any already?

The SL syllabus does not include integration by parts, I think. 

[thecsstudent]

A good advice my teacher gave us is, you can't always know which one to use but you should always try substitution first and if it doesn't work, try by parts. After a while of practice you will be able to tell intuitively which one to use in most cases.