What is the integral of (2sin(x)-5cos(x))/(cos(x)) ?

The integral of (2sin(x)-5cos(x))/(cos(x)) is -2ln|cos(x)|-5x+C

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integral of (2sin(x)-5cos(x))/(cos(x))

\int \frac{2 sin ( x ) - 5 cos ( x )}{cos ( x )} d x

- 2 ln ∣ cos (x) ∣ - 5 x + C

Solution steps

\int \frac{2 sin ( x ) - 5 cos ( x )}{cos ( x )} d x

Expand

\frac{2 sin ( x ) - 5 cos ( x )}{cos ( x )} : \frac{2 sin ( x )}{cos ( x )} - 5

= \int \frac{2 sin ( x )}{cos ( x )} - 5 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{2 sin ( x )}{cos ( x )} d x - \int 5 d x

\int \frac{2 sin ( x )}{cos ( x )} d x = - 2 ln ∣ cos (x) ∣

\int 5 d x = 5 x

= - 2 ln ∣ cos (x) ∣ - 5 x

Add a constant to the solution

= - 2 ln ∣ cos (x) ∣ - 5 x + C

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What is the integral of (2sin(x)-5cos(x))/(cos(x)) ?
The integral of (2sin(x)-5cos(x))/(cos(x)) is -2ln|cos(x)|-5x+C