With m an even integer and n an odd
integer; i.e.
Example 4: (much harder) I2 =
∫ tanm x secn x dx
I2 =
∫ tan2 x sec3
x dx
tan2x = sec2 x -
1
I2 =
∫ (1 + sec2 x) sec3 x dx
I2 =
∫ (sec5 x - sec3 x)dx Next use integration by parts
on each of these 2 integrals; first,
I2a = ∫sec3 x dx
Next let u
= sec x dv
= sec2x dx
du =
sec x tan x dx v
= tan x
I2a =
∫sec3 x dx =
sec x tan x - ∫sec x tan2 x dx
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