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

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

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

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

\frac{1}{1 + cot ( x )} + C

Solution steps

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

Apply u-substitution

: \int - \frac{1}{u ^{2}} d u

= \int - \frac{1}{u ^{2}} d u

Take the constant out:

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

= - \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= - (- \frac{1}{u})

Substitute back

u = 1 + cot (x)

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

Simplify

= \frac{1}{1 + cot ( x )}

Add a constant to the solution

= \frac{1}{1 + cot ( x )} + C

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