What is the integral of sec(10x) ?

The integral of sec(10x) is 1/10 ln|tan(10x)+sec(10x)|+C

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integral of sec(10x)

\int sec (10 x) d x

\frac{1}{10} ln ∣ tan (10 x) + sec (10 x) ∣ + C

Solution steps

\int sec (10 x) d x

Apply u-substitution

: \int sec (u) \frac{1}{10} d u

= \int sec (u) \frac{1}{10} d u

Take the constant out:

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

= \frac{1}{10} \cdot \int sec (u) d u

Use the common integral:

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

= \frac{1}{10} ln ∣ tan (u) + sec (u) ∣

Substitute back

u = 10 x

= \frac{1}{10} ln ∣ tan (10 x) + sec (10 x) ∣

Add a constant to the solution

= \frac{1}{10} ln ∣ tan (10 x) + sec (10 x) ∣ + C

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What is the integral of sec(10x) ?
The integral of sec(10x) is 1/10 ln|tan(10x)+sec(10x)|+C