What is the integral of csc^2(y) 1/(1+cot(y)) ?

The integral of csc^2(y) 1/(1+cot(y)) is -ln|1+cot(y)|+C

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integral of csc^2(y) 1/(1+cot(y))

\int csc^{2} (y) \frac{1}{1 + cot ( y )} d y

- ln ∣ 1 + cot (y) ∣ + C

Solution steps

\int csc^{2} (y) \frac{1}{1 + cot ( y )} d y

Simplify

csc^{2} (y) \frac{1}{1 + cot ( y )} : \frac{csc ^{2} ( y )}{1 + cot ( y )}

= \int \frac{csc ^{2} ( y )}{1 + cot ( y )} d y

Apply u-substitution

: \int - \frac{1}{u} d u

= \int - \frac{1}{u} d u

Take the constant out:

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

= - \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = 1 + cot (y)

= - ln ∣ 1 + cot (y) ∣

Add a constant to the solution

= - ln ∣ 1 + cot (y) ∣ + C

\frac{d}{d x} (\frac{x ^{2}}{y ^{3}})

x \to 2 lim (x^{2} + x + 2)

x \to 1 lim (sin (x))

r \to 9 lim (\frac{4 r - 3}{11})

x y^{^{'}} - 2 y = 5 x^{2}

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