What is the integral of (1+cos(x)+sin(x))/(cos(x)) ?

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

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integral of (1+cos(x)+sin(x))/(cos(x))

\int \frac{1 + cos ( x ) + sin ( x )}{cos ( x )} d x

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

Solution steps

\int \frac{1 + cos ( x ) + sin ( x )}{cos ( x )} d x

Expand

\frac{1 + cos ( x ) + sin ( x )}{cos ( x )} : \frac{1}{cos ( x )} + 1 + \frac{sin ( x )}{cos ( x )}

= \int \frac{1}{cos ( x )} + 1 + \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 \frac{1}{cos ( x )} d x + \int 1 d x + \int \frac{sin ( x )}{cos ( x )} d x

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

\int 1 d x = x

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

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

Add a constant to the solution

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

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