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

The integral of sqrt(cot(y))csc^2(y) is -2/3 cot^{3/2}(y)+C

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integral of sqrt(cot(y))csc^2(y)

\int cot (y) csc^{2} (y) d y

- \frac{2}{3} cot^{\frac{3}{2}} (y) + C

Solution steps

\int cot (y) csc^{2} (y) d y

Apply u-substitution

: \int - u d u

= \int - u d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= - \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = cot (y)

= - \frac{2}{3} cot^{\frac{3}{2}} (y)

Add a constant to the solution

= - \frac{2}{3} cot^{\frac{3}{2}} (y) + C

y^{^{''}} - 4 y^{^{'}} + 3 y = x

y^{^{'}} - \frac{6}{x} y = \frac{y ^{4}}{x ^{3}}

x \to 0 lim (\frac{arcsin ( x )}{7 x})

f (x) = sin^{4} (x), g (x) = sin^{2} (x)

t \to 0 lim (\frac{e ^{8 t} - 1}{t})

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