Am I allowed to do this?
for the integral of
∫ sec^4 (3x)/ tan^3 (3x) dx
I change it to
∫ sec^4 (3x) tan^-3 (3x)
From here I use the rule for trigonometry functions.
2 answers
Or do I use the rule of ∫ sec^n x dx divided by the rule of ∫ tan^n x dx?
If I go with my first assumption I get this:
∫ sec^4 (3x) tan^-3 (3x) = ∫ sec^3 (3x) tan^-4 (3x) sec(3x)tan(3x) dx
= ∫ sec^3 (3x) (sec^-4 3x - 1)sec(3x)tan(3x) dx
u = sec(3x) dx
du = 3sec(3x)tan(3x) dx > 1/3du
= 1/3 ∫ u^3 (u^-4 - 1) du
= 1/3 ∫ (u^-7 - u^3) du
= 1/3 (u^-6/-6 - u^4/4) + C
= -18/sec^6(3x) - sec^4 (3x)/12 + C
Would this be the right method and answer? Thank you
∫ sec^4 (3x) tan^-3 (3x) = ∫ sec^3 (3x) tan^-4 (3x) sec(3x)tan(3x) dx
= ∫ sec^3 (3x) (sec^-4 3x - 1)sec(3x)tan(3x) dx
u = sec(3x) dx
du = 3sec(3x)tan(3x) dx > 1/3du
= 1/3 ∫ u^3 (u^-4 - 1) du
= 1/3 ∫ (u^-7 - u^3) du
= 1/3 (u^-6/-6 - u^4/4) + C
= -18/sec^6(3x) - sec^4 (3x)/12 + C
Would this be the right method and answer? Thank you
1 answer