What is the integral of (-9+9sin(x))/(cos(x)) ?

The integral of (-9+9sin(x))/(cos(x)) is -9ln|tan(x)+sec(x)|-9ln|cos(x)|+C

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integral of (-9+9sin(x))/(cos(x))

\int \frac{- 9 + 9 sin ( x )}{cos ( x )} d x

- 9 ln ∣ tan (x) + sec (x) ∣ - 9 ln ∣ cos (x) ∣ + C

Solution steps

\int \frac{- 9 + 9 sin ( x )}{cos ( x )} d x

Expand

\frac{- 9 + 9 sin ( x )}{cos ( x )} : - \frac{9}{cos ( x )} + \frac{9 sin ( x )}{cos ( x )}

= \int - \frac{9}{cos ( x )} + \frac{9 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 \frac{9}{cos ( x )} d x + \int \frac{9 sin ( x )}{cos ( x )} d x

\int \frac{9}{cos ( x )} d x = 9 ln ∣ tan (x) + sec (x) ∣

\int \frac{9 sin ( x )}{cos ( x )} d x = - 9 ln ∣ cos (x) ∣

= - 9 ln ∣ tan (x) + sec (x) ∣ - 9 ln ∣ cos (x) ∣

Add a constant to the solution

= - 9 ln ∣ tan (x) + sec (x) ∣ - 9 ln ∣ cos (x) ∣ + C

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What is the integral of (-9+9sin(x))/(cos(x)) ?
The integral of (-9+9sin(x))/(cos(x)) is -9ln|tan(x)+sec(x)|-9ln|cos(x)|+C