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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{1}{u ( u + 1 )} d u

= \int \frac{1}{u ( u + 1 )} d u

Take the partial fraction of

\frac{1}{u ( u + 1 )} : \frac{1}{u} - \frac{1}{u + 1}

= \int \frac{1}{u} - \frac{1}{u + 1} 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

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

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{1}{u + 1} d u = ln ∣ u + 1 ∣

= ln ∣ u ∣ - ln ∣ u + 1 ∣

Substitute back

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

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

Add a constant to the solution

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

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