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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

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