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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

= sec (x) - ln ∣ tan (x) + sec (x) ∣

Add a constant to the solution

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

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