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

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

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

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

ln ∣ tan (x) + sec (x) ∣ - 2 sin (x) + C

Solution steps

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

Rewrite using trig identities

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

Expand

\frac{1 - 2 cos ^{2} ( x )}{cos ( x )} : \frac{1}{cos ( x )} - 2 cos (x)

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{1}{cos ( x )} d x = ln ∣ tan (x) + sec (x) ∣

\int 2 cos (x) d x = 2 sin (x)

= ln ∣ tan (x) + sec (x) ∣ - 2 sin (x)

Add a constant to the solution

= ln ∣ tan (x) + sec (x) ∣ - 2 sin (x) + C

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