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

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

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integral of 1/(1+sin(θ)+cos(θ))

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

ln tan (\frac{θ}{2}) + 1 + C

Solution steps

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

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{θ}{2}) + 1

Add a constant to the solution

= ln tan (\frac{θ}{2}) + 1 + C

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