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

The integral of (sin^2(x)-k)/(sin^2(x)) is x+kcot(x)+C

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

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

x + k cot (x) + C

Solution steps

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

Expand

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

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

\int 1 d x = x

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

= x - (- k cot (x))

Simplify

= x + k cot (x)

Add a constant to the solution

= x + k cot (x) + C

What is the integral of (sin^2(x)-k)/(sin^2(x)) ?
The integral of (sin^2(x)-k)/(sin^2(x)) is x+kcot(x)+C