What is the integral of (sin(x)+xcos(x))/(xsin(x)) ?

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

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integral of (sin(x)+xcos(x))/(xsin(x))

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

ln ∣ x ∣ + ln ∣ sin (x) ∣ + C

Solution steps

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

Expand

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ + ln ∣ sin (x) ∣

Add a constant to the solution

= ln ∣ x ∣ + ln ∣ sin (x) ∣ + C

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