What is the integral of sec(3x) ?

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

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integral of sec(3x)

\int sec (3 x) d x

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

Solution steps

\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

y^{^{'}} + 2 y = 7 x y^{2}

\int_{0}^{4} \frac{5 t}{( t - 5 ) ^{2}} d t

\int \frac{2}{( x - 1 ) ^{3}} d x

x \to 0 + lim (\frac{1}{e ^{x} - 1} - \frac{1}{x})

y = 4 - x^{2}, y = 0

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