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

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

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

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

- ln ∣ cos (x) ∣ - x + C

Solution steps

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

Expand

\frac{sin ( x ) - cos ( x )}{cos ( x )} : \frac{sin ( x )}{cos ( x )} - 1

= \int \frac{sin ( 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

= \int \frac{sin ( x )}{cos ( x )} d x - \int 1 d x

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

\int 1 d x = x

= - ln ∣ cos (x) ∣ - x

Add a constant to the solution

= - ln ∣ cos (x) ∣ - x + C

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\int \frac{arccos ( x )}{( 1 - x ^{2} ) ^{\frac{3}{2}}} d x

\int x^{2} - 3 x d x

\int \frac{1}{( x ^{2} + x ) ( x ^{2} + 9 )} d x

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