What is the integral of 1/(cos(3x)) ?

The integral of 1/(cos(3x)) is 1/3 ln|tan(3x)+sec(3x)|+C

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integral of 1/(cos(3x))

\int \frac{1}{cos ( 3 x )} d x

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

Solution steps

\int \frac{1}{cos ( 3 x )} d x

Rewrite using trig identities

= \int sec (3 x) d x

Apply u-substitution

: \int sec (u) \frac{1}{3} d u

= \int sec (u) \frac{1}{3} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int sec (u) d u

Use the common integral:

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= \frac{1}{3} ln ∣ tan (u) + sec (u) ∣

Substitute back

u = 3 x

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

Add a constant to the solution

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

- 9 x^{\frac{1}{2}} + x^{\frac{3}{2}}

(4, - 8), (8, 5)

\frac{d}{d x} (3 x^{4})

x \to 4 lim (\frac{4 - ∣ x ∣}{4 + x})

f (x) = \frac{1}{2} (7 t^{2} + t)^{- 3}

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