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

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

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

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

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

Solution steps

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

Apply u-substitution

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

Apply u-substitution

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

Take the constant out:

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

= - \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

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