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

y^{^{''}} - 2 y^{^{'}} + 2 y = e^{- t}, y (0) = 0, y^{^{'}} (0) = 1

x^{2 e^{- 42}}

\frac{9}{x}, 2 x, \frac{1}{2} x

\int 4 \frac{cos ( x ) + sin ( 2 x )}{sin ( x )} d x

y = cos^{2} (1 - 3 x^{2})

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