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

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

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integral of 3/(cos(3x))

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

ln ∣ tan (3 x) + sec (3 x) ∣ + C

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

= 3 \cdot \int sec (3 x) d x

Apply u-substitution

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

= 3 \cdot \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

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

Use the common integral:

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

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

Substitute back

u = 3 x

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

Simplify

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

= ln ∣ tan (3 x) + sec (3 x) ∣

Add a constant to the solution

= ln ∣ tan (3 x) + sec (3 x) ∣ + C

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