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

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

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

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

- ln ∣ 1 + cot (x) ∣ + C

Solution steps

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

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 (x)

= - ln ∣ 1 + cot (x) ∣

Add a constant to the solution

= - ln ∣ 1 + cot (x) ∣ + C

\int - e^{x} sin (y) d y

\frac{\partial}{\partial y} (x + 2 y + z)

\int x^{2} e^{- 1} d x

\int \frac{( x ^{2} - 4 x + 9 )}{x ^{2} - 6 x + 9} d x

(2 x + 3 y) d x + (3 x + 2) d y = 0

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