What is the integral of (1+tan(x))/(sec(x)) ?

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

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integral of (1+tan(x))/(sec(x))

\int \frac{1 + tan ( x )}{sec ( x )} d x

sin (x) - cos (x) + C

Solution steps

\int \frac{1 + tan ( x )}{sec ( x )} d x

Expand

\frac{1 + tan ( x )}{sec ( x )} : \frac{1}{sec ( x )} + \frac{tan ( x )}{sec ( x )}

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

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

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

= sin (x) - cos (x)

Add a constant to the solution

= sin (x) - cos (x) + C

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