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

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

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

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

2 1 - sin (x) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

= - \int - \frac{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}{u} d u)

Apply the Power Rule

= - (- 2 u^{\frac{1}{2}})

Substitute back

u = 1 - sin (x)

= - (- 2 (1 - sin (x))^{\frac{1}{2}})

Simplify

- (- 2 (1 - sin (x))^{\frac{1}{2}}) : 2 1 - sin (x)

= 2 1 - sin (x)

Add a constant to the solution

= 2 1 - sin (x) + C

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