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.
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
To evaluate the integral ∫ sec^4 (3x)/ tan^3 (3x) dx, the first step you took was to change the exponent of tan from positive to negative. This is a valid transformation because you are essentially moving tan^3 (3x) from the numerator to the denominator.
Now, you can apply the rule for trigonometric functions, specifically the one that states:
tan^-n (x) = (1/tan^n (x))
Using this rule, you can rewrite the integral as:
∫ sec^4 (3x) / (1/tan^3 (3x)) dx
Next, you can simplify the expression further by multiplying sec^4 (3x) with tan^3 (3x):
∫ (sec^4 (3x) * tan^3 (3x)) dx
Now, you have transformed the integral into a simpler form that can be evaluated more easily.