What is the integral of (sin(7x))/(sqrt(cos(7x)-2)) ?

The integral of (sin(7x))/(sqrt(cos(7x)-2)) is -2/7 sqrt(cos(7x)-2)+C

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integral of (sin(7x))/(sqrt(cos(7x)-2))

\int \frac{sin ( 7 x )}{cos ( 7 x ) - 2} d x

- \frac{2}{7} cos (7 x) - 2 + C

Solution steps

\int \frac{sin ( 7 x )}{cos ( 7 x ) - 2} d x

Apply u-substitution

: \int - \frac{1}{7 u} d u

= \int - \frac{1}{7 u} d u

Take the constant out:

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

= - \frac{1}{7} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

= - \frac{1}{7} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = cos (7 x) - 2

= - \frac{1}{7} \cdot 2 (cos (7 x) - 2)^{\frac{1}{2}}

Simplify

- \frac{1}{7} \cdot 2 (cos (7 x) - 2)^{\frac{1}{2}} : - \frac{2}{7} cos (7 x) - 2

= - \frac{2}{7} cos (7 x) - 2

Add a constant to the solution

= - \frac{2}{7} cos (7 x) - 2 + C

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What is the integral of (sin(7x))/(sqrt(cos(7x)-2)) ?
The integral of (sin(7x))/(sqrt(cos(7x)-2)) is -2/7 sqrt(cos(7x)-2)+C