What is the integral of 1/(cos^2(x)-sin^2(x)) ?

The integral of 1/(cos^2(x)-sin^2(x)) is 1/2 ln|tan(2x)+sec(2x)|+C

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integral of 1/(cos^2(x)-sin^2(x))

\int \frac{1}{cos ^{2} ( x ) - sin ^{2} ( x )} d x

\frac{1}{2} ln ∣ tan (2 x) + sec (2 x) ∣ + C

Solution steps

\int \frac{1}{cos ^{2} ( x ) - sin ^{2} ( x )} d x

Rewrite using trig identities

= \int sec (2 x) d x

Apply u-substitution

: \int sec (u) \frac{1}{2} d u

= \int sec (u) \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int sec (u) d u

Use the common integral:

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

= \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

Substitute back

u = 2 x

= \frac{1}{2} ln ∣ tan (2 x) + sec (2 x) ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ tan (2 x) + sec (2 x) ∣ + C

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What is the integral of 1/(cos^2(x)-sin^2(x)) ?
The integral of 1/(cos^2(x)-sin^2(x)) is 1/2 ln|tan(2x)+sec(2x)|+C