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

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

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

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

ln ∣ sin (x) ∣ + ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

= ln ∣ sin (x) ∣ + ln ∣ x ∣

Add a constant to the solution

= ln ∣ sin (x) ∣ + ln ∣ x ∣ + C

\int (\frac{tan ( x )}{cos ( x )} - csc^{2} (x)) d x

\int \frac{1}{4 ^{x} - 4} d x

\int \frac{4 x}{2 x + 3} d x

\int 4 e^{y} d y

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