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

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

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

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

- 2 (1 + cos (x) - ln 1 + cos (x)) + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{2 ( v - 1 )}{v} d v

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

Take the constant out:

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

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

Expand

\frac{v - 1}{v} : 1 - \frac{1}{v}

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

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

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= - 2 (v - ln ∣ v ∣)

Substitute back

= - 2 (1 + cos (x) - ln 1 + cos (x))

Add a constant to the solution

= - 2 (1 + cos (x) - ln 1 + cos (x)) + C

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