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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

Expand

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

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

\int 2 sin (x) d x = - 2 cos (x)

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

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

Add a constant to the solution

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

\int \frac{1}{x} (1 + ln (x)) d x

\int \frac{x ^{2} - 1}{x ^{2} + 4} d x

\int \frac{1}{x + c} d x

\int e^{y} cot^{2} (e^{y} + 1) d y

\int cot^{2} (\frac{4 x}{π}) d x

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