What is the integral of 1/(cos(4θ)) ?

The integral of 1/(cos(4θ)) is 1/4 ln|tan(4θ)+sec(4θ)|+C

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integral of 1/(cos(4θ))

\int \frac{1}{cos ( 4 θ )} d θ

\frac{1}{4} ln ∣ tan (4 θ) + sec (4 θ) ∣ + C

Solution steps

\int \frac{1}{cos ( 4 θ )} d θ

Rewrite using trig identities

= \int sec (4 θ) d θ

Apply u-substitution

: \int sec (u) \frac{1}{4} d u

= \int sec (u) \frac{1}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int sec (u) d u

Use the common integral:

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

= \frac{1}{4} ln ∣ tan (u) + sec (u) ∣

Substitute back

u = 4 θ

= \frac{1}{4} ln ∣ tan (4 θ) + sec (4 θ) ∣

Add a constant to the solution

= \frac{1}{4} ln ∣ tan (4 θ) + sec (4 θ) ∣ + C

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What is the integral of 1/(cos(4θ)) ?
The integral of 1/(cos(4θ)) is 1/4 ln|tan(4θ)+sec(4θ)|+C