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

The integral of (2-sqrt(cot(x)))^2csc^2(x) is -4cot(x)+8/3 cot^{3/2}(x)-1/2 csc^2(x)+C

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

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

- 4 cot (x) + \frac{8}{3} cot^{\frac{3}{2}} (x) - \frac{1}{2} csc^{2} (x) + C

Solution steps

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

Rewrite using trig identities

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

Expand

\frac{( 2 - cot ( x ) ) ^{2}}{sin ^{2} ( x )} : \frac{4}{sin ^{2} ( x )} - \frac{4 cot ( x )}{sin ^{2} ( x )} + \frac{cot ( x )}{sin ^{2} ( x )}

= \int \frac{4}{sin ^{2} ( x )} - \frac{4 cot ( x )}{sin ^{2} ( x )} + \frac{cot ( 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{4}{sin ^{2} ( x )} d x - \int \frac{4 cot ( x )}{sin ^{2} ( x )} d x + \int \frac{cot ( x )}{sin ^{2} ( x )} d x

\int \frac{4}{sin ^{2} ( x )} d x = - 4 cot (x)

\int \frac{4 cot ( x )}{sin ^{2} ( x )} d x = - \frac{8}{3} cot^{\frac{3}{2}} (x)

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

= - 4 cot (x) - (- \frac{8}{3} cot^{\frac{3}{2}} (x)) - \frac{1}{2} csc^{2} (x)

Simplify

= - 4 cot (x) + \frac{8}{3} cot^{\frac{3}{2}} (x) - \frac{1}{2} csc^{2} (x)

Add a constant to the solution

= - 4 cot (x) + \frac{8}{3} cot^{\frac{3}{2}} (x) - \frac{1}{2} csc^{2} (x) + C

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What is the integral of (2-sqrt(cot(x)))^2csc^2(x) ?
The integral of (2-sqrt(cot(x)))^2csc^2(x) is -4cot(x)+8/3 cot^{3/2}(x)-1/2 csc^2(x)+C