What is the integral of (1-cot(x))csc^2(x) ?

The integral of (1-cot(x))csc^2(x) is -cot(x)+1/2 csc^2(x)+C

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integral of (1-cot(x))csc^2(x)

\int (1 - cot (x)) csc^{2} (x) d x

- cot (x) + \frac{1}{2} csc^{2} (x) + C

Solution steps

\int (1 - cot (x)) csc^{2} (x) d x

Rewrite using trig identities

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

Expand

\frac{sin ( x ) - cos ( x )}{sin ^{3} ( x )} : \frac{1}{sin ^{2} ( x )} - \frac{cos ( x )}{sin ^{3} ( x )}

= \int \frac{1}{sin ^{2} ( x )} - \frac{cos ( x )}{sin ^{3} ( 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 ^{2} ( x )} d x - \int \frac{cos ( x )}{sin ^{3} ( x )} d x

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

\int \frac{cos ( x )}{sin ^{3} ( x )} d x = - \frac{1}{2} csc^{2} (x)

= - cot (x) - (- \frac{1}{2} csc^{2} (x))

Simplify

= - cot (x) + \frac{1}{2} csc^{2} (x)

Add a constant to the solution

= - cot (x) + \frac{1}{2} csc^{2} (x) + C

\frac{d}{d θ} (- 6 sin (θ))

\frac{d}{d x} (- \frac{1}{( 2 x + 8 ) ^{\frac{3}{2}}})

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

\frac{d}{d x} (\frac{x}{x + 5})

\frac{d}{d t} (cos (2 t))

What is the integral of (1-cot(x))csc^2(x) ?
The integral of (1-cot(x))csc^2(x) is -cot(x)+1/2 csc^2(x)+C