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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{1 + u ^{2}}{2 u} d u

= \int \frac{1 + u ^{2}}{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{1 + u ^{2}}{u} d u

Apply u-substitution

: \int \frac{v ^{2}}{v ^{2} - 1} d v

= \frac{1}{2} \cdot \int \frac{v ^{2}}{v ^{2} - 1} d v

Factor

v^{2} - 1 : - (- v^{2} + 1)

= \frac{1}{2} \cdot \int \frac{v ^{2}}{- ( - v ^{2} + 1 )} d v

Take the constant out:

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

= \frac{1}{2} (- \int \frac{v ^{2}}{- v ^{2} + 1} d v)

\frac{v ^{2}}{- v ^{2} + 1} = \frac{1}{- v ^{2} + 1} - 1

= \frac{1}{2} (- \int \frac{1}{- v ^{2} + 1} - 1 d v)

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 \frac{1}{- v ^{2} + 1} d v - \int 1 d v))

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

\int 1 d v = v

= \frac{1}{2} (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2} - v))

Substitute back

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

Simplify

\frac{1}{2} - \frac{ln 1 + tan ^{2} ( \frac{x}{2} ) + 1}{2} - \frac{ln 1 + tan ^{2} ( \frac{x}{2} ) - 1}{2} - 1 + tan^{2} (\frac{x}{2}) : - \frac{1}{2 2} (ln sec (\frac{x}{2}) + 1 - ln sec (\frac{x}{2}) - 1 - 2 sec (\frac{x}{2}))

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

Add a constant to the solution

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

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