What is the integral of sin(x)(tan(x)-cot(x))^2 ?

The integral of sin(x)(tan(x)-cot(x))^2 is 4cos(x)+sec(x)+ln|tan(x/2)|+C

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integral of sin(x)(tan(x)-cot(x))^2

\int sin (x) (tan (x) - cot (x))^{2} d x

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

Solution steps

\int sin (x) (tan (x) - cot (x))^{2} d x

Rewrite using trig identities

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

Expand

\frac{( sin ^{2} ( x ) - cos ^{2} ( x ) ) ^{2}}{cos ^{2} ( x ) sin ( x )} : \frac{sin ^{3} ( x )}{cos ^{2} ( x )} - 2 sin (x) + \frac{cos ^{2} ( x )}{sin ( x )}

= \int \frac{sin ^{3} ( x )}{cos ^{2} ( x )} - 2 sin (x) + \frac{cos ^{2} ( 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{sin ^{3} ( x )}{cos ^{2} ( x )} d x - \int 2 sin (x) d x + \int \frac{cos ^{2} ( x )}{sin ( x )} d x

\int \frac{sin ^{3} ( x )}{cos ^{2} ( x )} d x = cos (x) + sec (x)

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

\int \frac{cos ^{2} ( x )}{sin ( x )} d x = ln tan (\frac{x}{2}) + cos (x)

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

Simplify

= 4 cos (x) + sec (x) + ln tan (\frac{x}{2})

Add a constant to the solution

= 4 cos (x) + sec (x) + ln tan (\frac{x}{2}) + C

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What is the integral of sin(x)(tan(x)-cot(x))^2 ?
The integral of sin(x)(tan(x)-cot(x))^2 is 4cos(x)+sec(x)+ln|tan(x/2)|+C