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

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

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integral of (cos(x)-xsin(x))/(xcos(x))

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

ln ∣ x ∣ + ln ∣ cos (x) ∣ + C

Solution steps

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

Expand

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ - (- ln ∣ cos (x) ∣)

Simplify

= ln ∣ x ∣ + ln ∣ cos (x) ∣

Add a constant to the solution

= ln ∣ x ∣ + ln ∣ cos (x) ∣ + C

\frac{d}{d x} (ln (\frac{x + x ^{2} + 4}{2}))

f (x) = e^{x} - x - 2

\frac{\partial}{\partial x} (e^{x} + 3 e^{y})

\frac{d y}{d x}, x y = x^{5} y + 34

\frac{d}{d x} (\frac{3}{sin ( x )})

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