What is the integral of (sec(7x))/7 ?

The integral of (sec(7x))/7 is 1/49 ln|tan(7x)+sec(7x)|+C

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integral of (sec(7x))/7

\int \frac{sec ( 7 x )}{7} d x

\frac{1}{49} ln ∣ tan (7 x) + sec (7 x) ∣ + C

Solution steps

\int \frac{sec ( 7 x )}{7} d x

Take the constant out:

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

= \frac{1}{7} \cdot \int sec (7 x) d x

Apply u-substitution

= \frac{1}{7} \cdot \int sec (u) \frac{1}{7} d u

Take the constant out:

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

= \frac{1}{7} \cdot \frac{1}{7} \cdot \int sec (u) d u

Use the common integral:

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

= \frac{1}{7} \cdot \frac{1}{7} ln ∣ tan (u) + sec (u) ∣

Substitute back

u = 7 x

= \frac{1}{7} \cdot \frac{1}{7} ln ∣ tan (7 x) + sec (7 x) ∣

Simplify

\frac{1}{7} \cdot \frac{1}{7} ln ∣ tan (7 x) + sec (7 x) ∣ : \frac{1}{49} ln ∣ tan (7 x) + sec (7 x) ∣

= \frac{1}{49} ln ∣ tan (7 x) + sec (7 x) ∣

Add a constant to the solution

= \frac{1}{49} ln ∣ tan (7 x) + sec (7 x) ∣ + C

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What is the integral of (sec(7x))/7 ?
The integral of (sec(7x))/7 is 1/49 ln|tan(7x)+sec(7x)|+C