What is the integral of (cos(5x))/(sqrt(1-sin(5x))) ?

The integral of (cos(5x))/(sqrt(1-sin(5x))) is -2/5 sqrt(1-sin(5x))+C

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integral of (cos(5x))/(sqrt(1-sin(5x)))

\int \frac{cos ( 5 x )}{1 - sin ( 5 x )} d x

- \frac{2}{5} 1 - sin (5 x) + C

Solution steps

\int \frac{cos ( 5 x )}{1 - sin ( 5 x )} d x

Apply u-substitution

: \int \frac{cos ( u )}{5 1 - sin ( u )} d u

= \int \frac{cos ( u )}{5 1 - sin ( u )} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{cos ( u )}{1 - sin ( u )} d u

Apply u-substitution

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

= \frac{1}{5} \cdot \int - \frac{1}{v} d v

Take the constant out:

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

= \frac{1}{5} (- \int \frac{1}{v} d v)

Apply the Power Rule

: 2 v^{\frac{1}{2}}

= \frac{1}{5} (- 2 v^{\frac{1}{2}})

Substitute back

= \frac{1}{5} (- 2 (1 - sin (5 x))^{\frac{1}{2}})

Simplify

\frac{1}{5} (- 2 (1 - sin (5 x))^{\frac{1}{2}}) : - \frac{2}{5} 1 - sin (5 x)

= - \frac{2}{5} 1 - sin (5 x)

Add a constant to the solution

= - \frac{2}{5} 1 - sin (5 x) + C

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What is the integral of (cos(5x))/(sqrt(1-sin(5x))) ?
The integral of (cos(5x))/(sqrt(1-sin(5x))) is -2/5 sqrt(1-sin(5x))+C