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

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

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

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

sin (x) + ln ∣ cos (x) ∣ + C

Solution steps

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

Expand

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

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

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

\int \frac{sin ( x )}{cos ( x )} d x = - ln ∣ cos (x) ∣

= sin (x) - (- ln ∣ cos (x) ∣)

Simplify

= sin (x) + ln ∣ cos (x) ∣

Add a constant to the solution

= sin (x) + ln ∣ cos (x) ∣ + C

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