What is the integral of sec(3x+5) ?

The integral of sec(3x+5) is 1/3 ln|tan(3x+5)+sec(3x+5)|+C

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integral of sec(3x+5)

\int sec (3 x + 5) d x

\frac{1}{3} ln ∣ tan (3 x + 5) + sec (3 x + 5) ∣ + C

Solution steps

\int sec (3 x + 5) 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 + 5

= \frac{1}{3} ln ∣ tan (3 x + 5) + sec (3 x + 5) ∣

Add a constant to the solution

= \frac{1}{3} ln ∣ tan (3 x + 5) + sec (3 x + 5) ∣ + C

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What is the integral of sec(3x+5) ?
The integral of sec(3x+5) is 1/3 ln|tan(3x+5)+sec(3x+5)|+C