What is the integral of (sin(x)-cos(x))/(sin^2(x)) ?

The integral of (sin(x)-cos(x))/(sin^2(x)) is ln|tan(x/2)|+csc(x)+C

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integral of (sin(x)-cos(x))/(sin^2(x))

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

ln tan (\frac{x}{2}) + csc (x) + C

Solution steps

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

Expand

\frac{sin ( x ) - cos ( x )}{sin ^{2} ( x )} : \frac{1}{sin ( x )} - \frac{cos ( x )}{sin ^{2} ( x )}

= \int \frac{1}{sin ( x )} - \frac{cos ( x )}{sin ^{2} ( 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}{sin ( x )} d x - \int \frac{cos ( x )}{sin ^{2} ( x )} d x

\int \frac{1}{sin ( x )} d x = ln tan (\frac{x}{2})

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

= ln tan (\frac{x}{2}) - (- csc (x))

Simplify

= ln tan (\frac{x}{2}) + csc (x)

Add a constant to the solution

= ln tan (\frac{x}{2}) + csc (x) + C

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What is the integral of (sin(x)-cos(x))/(sin^2(x)) ?
The integral of (sin(x)-cos(x))/(sin^2(x)) is ln|tan(x/2)|+csc(x)+C