What is ln|tan(3x)+sec(3x)|+1/3 ln|cos(3x)|+C ?

The solution to ln|tan(3x)+sec(3x)|+1/3 ln|cos(3x)|+C is ln(|tan(3x)+sec(3x)||cos(3x)|^{1/3})+C

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simplify ln|tan(3x)+sec(3x)|+1/3 ln|cos(3x)|+C

simplify

ln ∣ tan (3 x) + sec (3 x) ∣ + \frac{1}{3} ln ∣ cos (3 x) ∣ + C

ln (∣ tan (3 x) + sec (3 x) ∣ ∣ cos (3 x) ∣^{\frac{1}{3}}) + C

Solution steps

ln ∣ tan (3 x) + sec (3 x) ∣ + \frac{1}{3} ln ∣ cos (3 x) ∣ + C

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{3} ln ∣ cos (3 x) ∣ = ln (∣ cos (3 x) ∣^{\frac{1}{3}})

= ln ∣ tan (3 x) + sec (3 x) ∣ + ln (∣ cos (3 x) ∣^{\frac{1}{3}}) + C

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

ln ∣ tan (3 x) + sec (3 x) ∣ + ln (∣ cos (3 x) ∣^{\frac{1}{3}}) = ln (∣ tan (3 x) + sec (3 x) ∣ ∣ cos (3 x) ∣^{\frac{1}{3}})

= ln (∣ tan (3 x) + sec (3 x) ∣ ∣ cos (3 x) ∣^{\frac{1}{3}}) + C

lo g_{e} (\frac{3}{5})

lo g_{2} ((x^{3} + 2) (x - 2)^{3})

- \frac{1}{0.024} ln (\frac{6 - 1}{4.2})

(1 - x) (\frac{1}{1 - x ^{2}}) - \frac{1}{2} ln (\frac{1 + x}{1 - x})

- lo g_{10} (4.88 \cdot 1 0^{- 6})

What is ln|tan(3x)+sec(3x)|+1/3 ln|cos(3x)|+C ?
The solution to ln|tan(3x)+sec(3x)|+1/3 ln|cos(3x)|+C is ln(|tan(3x)+sec(3x)||cos(3x)|^{1/3})+C