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)+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}) + 1 + C

Solution steps

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

Apply u-substitution

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

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

Apply u-substitution

: \int \frac{1}{v} d v

= \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= ln ∣ v ∣

Substitute back

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

Add a constant to the solution

= 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)+1|+C