What is the integral of (wcos(w)+sin(w))/(wsin(w)) ?

The integral of (wcos(w)+sin(w))/(wsin(w)) is ln|sin(w)|+ln|w|+C

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integral of (wcos(w)+sin(w))/(wsin(w))

\int \frac{w cos ( w ) + sin ( w )}{w sin ( w )} d w

ln ∣ sin (w) ∣ + ln ∣ w ∣ + C

Solution steps

\int \frac{w cos ( w ) + sin ( w )}{w sin ( w )} d w

Expand

\frac{w cos ( w ) + sin ( w )}{w sin ( w )} : \frac{cos ( w )}{sin ( w )} + \frac{1}{w}

= \int \frac{cos ( w )}{sin ( w )} + \frac{1}{w} d w

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{cos ( w )}{sin ( w )} d w + \int \frac{1}{w} d w

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

\int \frac{1}{w} d w = ln ∣ w ∣

= ln ∣ sin (w) ∣ + ln ∣ w ∣

Add a constant to the solution

= ln ∣ sin (w) ∣ + ln ∣ w ∣ + C

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What is the integral of (wcos(w)+sin(w))/(wsin(w)) ?
The integral of (wcos(w)+sin(w))/(wsin(w)) is ln|sin(w)|+ln|w|+C