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What is # int \ tan3xsec3x \ dx#?

Answer 1

Christopher Allers

# int \ tan3xsec3x \ dx = 1/3sec3x + C#

A standard trigonometry differential is:

# d/dx sec x = secxtanx iff int \ secxtanx \ dx = sec x \ \ (+C) #

We can see a very close similarity with this result and out integral, so note that:

# d/dx sec ax = asecaxtanax #

for constant #a#, Hence we have:

# int \ tan3xsec3x \ dx = 1/3sec3x + C#

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Answer 2

Paisley Johnson

To find the integral of ( \int \tan(3x) \sec(3x) , dx ), we use the substitution method. Let ( u = 3x ), then ( du = 3 , dx ). Rewrite the integral using ( u ):

[ \int \tan(u) \sec(u) \frac{1}{3} , du ]

[ = \frac{1}{3} \int \tan(u) \sec(u) , du ]

Now, use integration by parts with ( u = \tan(u) ) and ( dv = \sec(u) , du ). This gives:

[ = \frac{1}{3} (\tan(u) \cdot \ln|\sec(u) + \tan(u)| - \int \ln|\sec(u) + \tan(u)| , du) ]

The integral ( \int \ln|\sec(u) + \tan(u)| , du ) does not have a simple closed-form antiderivative. Thus, the integral ( \int \tan(3x) \sec(3x) , dx ) cannot be expressed in elementary functions.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.