What is the integral of (cos(8x))/(1-cos(8x)) ?

The integral of (cos(8x))/(1-cos(8x)) is -x-1/8 cot(4x)+C

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integral of (cos(8x))/(1-cos(8x))

\int \frac{cos ( 8 x )}{1 - cos ( 8 x )} d x

- x - \frac{1}{8} cot (4 x) + C

Solution steps

\int \frac{cos ( 8 x )}{1 - cos ( 8 x )} d x

Long division

\frac{cos ( 8 x )}{1 - cos ( 8 x )} : - 1 + \frac{1}{- cos ( 8 x ) + 1}

= \int - 1 + \frac{1}{- cos ( 8 x ) + 1} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int 1 d x + \int \frac{1}{- cos ( 8 x ) + 1} d x

\int 1 d x = x

\int \frac{1}{- cos ( 8 x ) + 1} d x = - \frac{1}{8} cot (4 x)

= - x - \frac{1}{8} cot (4 x)

Add a constant to the solution

= - x - \frac{1}{8} cot (4 x) + C

\int 0.03 e^{0.05 x} d x

f (x) = \frac{9 ( 5 ^{x} )}{x ^{5}}

\int lo g_{x} (x) d x

\int \frac{5 x ^{2} - 3}{x ( x ^{2} - 1 )} d x

f (x) = \frac{4 x ^{2} + 8 x + 8}{x}

What is the integral of (cos(8x))/(1-cos(8x)) ?
The integral of (cos(8x))/(1-cos(8x)) is -x-1/8 cot(4x)+C