What is the integral of (sec(x)-2tan(x))/(3cos(x)) ?

The integral of (sec(x)-2tan(x))/(3cos(x)) is 1/3 (tan(x)-2sec(x))+C

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integral of (sec(x)-2tan(x))/(3cos(x))

\int \frac{sec ( x ) - 2 tan ( x )}{3 cos ( x )} d x

\frac{1}{3} (tan (x) - 2 sec (x)) + C

Solution steps

\int \frac{sec ( x ) - 2 tan ( x )}{3 cos ( x )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int \frac{sec ( x ) - 2 tan ( x )}{cos ( x )} d x

Expand

\frac{sec ( x ) - 2 tan ( x )}{cos ( x )} : \frac{sec ( x )}{cos ( x )} - \frac{2 tan ( x )}{cos ( x )}

= \frac{1}{3} \cdot \int \frac{sec ( x )}{cos ( x )} - \frac{2 tan ( 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

= \frac{1}{3} (\int \frac{sec ( x )}{cos ( x )} d x - \int \frac{2 tan ( x )}{cos ( x )} d x)

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

\int \frac{2 tan ( x )}{cos ( x )} d x = 2 sec (x)

= \frac{1}{3} (tan (x) - 2 sec (x))

Add a constant to the solution

= \frac{1}{3} (tan (x) - 2 sec (x)) + C

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What is the integral of (sec(x)-2tan(x))/(3cos(x)) ?
The integral of (sec(x)-2tan(x))/(3cos(x)) is 1/3 (tan(x)-2sec(x))+C