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

\int_{0}^{\infty} 6 sin^{2} (x) d x

x \to - 1 lim (2 x + 6)

\int (\frac{1}{x ^{11}}) d x

\int csc^{3} (2 x + 1) d x

(2.3) (8.6)

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