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

The integral of (sec(x)-1/(sin(x))+cos(x)) is ln|tan(x)+sec(x)|-ln|tan(x/2)|+sin(x)+C

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

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

ln ∣ tan (x) + sec (x) ∣ - ln tan (\frac{x}{2}) + sin (x) + C

Solution steps

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

\int sec (x) d x = ln ∣ tan (x) + sec (x) ∣

\int \frac{1}{sin ( x )} d x = ln tan (\frac{x}{2})

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

= ln ∣ tan (x) + sec (x) ∣ - ln tan (\frac{x}{2}) + sin (x)

Add a constant to the solution

= ln ∣ tan (x) + sec (x) ∣ - ln tan (\frac{x}{2}) + sin (x) + C

\frac{\partial}{\partial x} (sin (\frac{x}{1 + y}))

\int 27 sec^{- 2} (x t a) n^{3} x d x

\frac{d y}{d x} = y (c_{1} + π^{2}) (1 + x)^{- 1}

\frac{d}{d x} (1 - cos (x))

What is the integral of (sec(x)-1/(sin(x))+cos(x)) ?
The integral of (sec(x)-1/(sin(x))+cos(x)) is ln|tan(x)+sec(x)|-ln|tan(x/2)|+sin(x)+C