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

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

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integral of 1/(csc(5x)sqrt(cos(5x)))

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

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

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

Apply u-substitution

= \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

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

Substitute back

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

Simplify

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

= - \frac{2}{5} cos (5 x)

Add a constant to the solution

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

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