He hasn't exactly given the answer away - he's just pointed out what an appropriate substitution would be.

Tbh I'd expect the OP's textbook to give guidance on what substitution is appropriate when integrating a rational function of sin and cos, but the OP has probably forgotten this :smile:

Reply 10

tiny hobbit

Original post by Mr M

I'm sorely tempted to create a sticky about the spelling of 'integration'.

Reply 11

newblood

For the first integral why dont you rewrite it in terms of cos and then see what multiplying top and bottom by cosx will give you.

Reply 12

TeeEm

Original post by newblood

For the first integral why dont you rewrite it in terms of cos and then see what multiplying top and bottom by cosx will give you.

Your suggestion will work, so so will the substitution t = tan(tx/2) which I suggested for the second integral.

However the best method to integrate secx is to write it as secx/1, multiply top and bottom by [secx + tan], expand and simplify and then notice that numerator is the exact derivative of denominator.

I think our friend asks for the second integral only

Reply 13

TeeEm

Original post by kabilan13

integrate (1/(sect-1))

method A, Standard way, using little t identities

use the substitution u=tan(t/2)
then sect = (1+u²)/(1-u²)

Method B
multiply top and bottom by (sect+1)
bottom gives a very well known trig identity
you split the fraction and tidy up ... bit messy
eventually after a bit of work you get cot t cosect + cot²t
the first "lump" is an exact differential the second can be switched into cosec²t - 1, which also integrates.
There is a problem with this method if your limits cross the discontinuity, i.e. sect + 1 which we multiplied top and bottom must not be zero.

METHOD C
multiply top and bottom by cost first

write the cost in the top, as cos²(t/2) - sin²(t/2)
write the cost in the bottom as 2cos²(t/2) - 1
Split fraction to 1/2 -1/2tan2(t/2)
etc etc...
Again potential problem in definite integration as in method B.

I wish I could point you to download my "mother of all resources" when it comes to integration but I do not want to get done for advertising.