What is the integral of (cot(x))/(sin(x)+cos(x)+1) ?

The integral of (cot(x))/(sin(x)+cos(x)+1) is -1/2 (tan(x/2)-ln|tan(x/2)|)+C

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integral of (cot(x))/(sin(x)+cos(x)+1)

\int \frac{cot ( x )}{sin ( x ) + cos ( x ) + 1} d x

- \frac{1}{2} (tan (\frac{x}{2}) - ln tan (\frac{x}{2})) + C

Solution steps

\int \frac{cot ( x )}{sin ( x ) + cos ( x ) + 1} d x

Apply u-substitution

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

Take the constant out:

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

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

Expand

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \frac{1}{2} (\int 1 d u - \int \frac{1}{u} d u)

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= - \frac{1}{2} (u - ln ∣ u ∣)

Substitute back

u = tan (\frac{x}{2})

= - \frac{1}{2} (tan (\frac{x}{2}) - ln tan (\frac{x}{2}))

Add a constant to the solution

= - \frac{1}{2} (tan (\frac{x}{2}) - ln tan (\frac{x}{2})) + C

What is the integral of (cot(x))/(sin(x)+cos(x)+1) ?
The integral of (cot(x))/(sin(x)+cos(x)+1) is -1/2 (tan(x/2)-ln|tan(x/2)|)+C