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

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

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

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

- ln ∣ cot (x) + 1 ∣ + C

Solution steps

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

= - ln ∣ cot (x) + 1 ∣

Add a constant to the solution

= - ln ∣ cot (x) + 1 ∣ + C

f (x) = (2 + 5 x) (5 - 4 x), at x = 1

y^{^{'}} - y = 5.2

\int sin (3 z) cos (2) (3 z) d z

f (t) = t cos (2 t) e^{- t}, at t = 3

(1 + x^{2} + y^{2} + x^{2} y^{2}) d y = y^{2} d x

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