What is the integral of sec(3t) ?

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

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integral of sec(3t)

\int sec (3 t) d t

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

Solution steps

\int sec (3 t) d t

Apply u-substitution

= \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 t

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

Add a constant to the solution

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

x \to - 1 lim (5)

\frac{d}{d x} (x^{\frac{1}{2}} + 4 x^{\frac{2}{3}} + 7 x + 3)

x \to 0 lim (cot (7 x))

\int 9 tan (9 x) d x

\frac{d}{d m} (5 m^{2} + 15 m^{3})

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