What is the integral of (sec(x)+cos(x))/(5cos(x)) ?

The integral of (sec(x)+cos(x))/(5cos(x)) is 1/5 (tan(x)+x)+C

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integral of (sec(x)+cos(x))/(5cos(x))

\int \frac{sec ( x ) + cos ( x )}{5 cos ( x )} d x

\frac{1}{5} (tan (x) + x) + C

Solution steps

\int \frac{sec ( x ) + cos ( x )}{5 cos ( x )} d x

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{sec ( x ) + cos ( x )}{cos ( x )} d x

Expand

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

= \frac{1}{5} \cdot \int \frac{sec ( x )}{cos ( x )} + 1 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}{5} (\int \frac{sec ( x )}{cos ( x )} d x + \int 1 d x)

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

\int 1 d x = x

= \frac{1}{5} (tan (x) + x)

Add a constant to the solution

= \frac{1}{5} (tan (x) + x) + C

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What is the integral of (sec(x)+cos(x))/(5cos(x)) ?
The integral of (sec(x)+cos(x))/(5cos(x)) is 1/5 (tan(x)+x)+C