What is the integral of (32)/(1-cos(8x)) ?

The integral of (32)/(1-cos(8x)) is -4cot(4x)+C

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integral of (32)/(1-cos(8x))

\int \frac{32}{1 - cos ( 8 x )} d x

- 4 cot (4 x) + C

Solution steps

\int \frac{32}{1 - cos ( 8 x )} d x

Take the constant out:

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

= 32 \cdot \int \frac{1}{1 - cos ( 8 x )} d x

Apply u-substitution

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

= 32 \cdot \int \frac{1}{8 ( 1 - cos ( u ) )} d u

Take the constant out:

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

= 32 \cdot \frac{1}{8} \cdot \int \frac{1}{1 - cos ( u )} d u

Apply u-substitution

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

= 32 \cdot \frac{1}{8} \cdot \int \frac{1}{v ^{2}} d v

Apply the Power Rule

: - \frac{1}{v}

= 32 \cdot \frac{1}{8} (- \frac{1}{v})

Substitute back

= 32 \cdot \frac{1}{8} (- \frac{1}{tan ( \frac{8 x}{2} )})

Simplify

32 \cdot \frac{1}{8} (- \frac{1}{tan ( \frac{8 x}{2} )}) : - 4 cot (4 x)

= - 4 cot (4 x)

Add a constant to the solution

= - 4 cot (4 x) + C

\int \frac{1}{4} e^{2 x} d x

\frac{d}{d x} (\frac{1}{2 x} + 2 x)

\int_{4}^{9} \frac{1}{7 - x} d x

\frac{d y}{d x} = e^{6 x} + 12 y

(\frac{1}{2 x + 3})^{3}

What is the integral of (32)/(1-cos(8x)) ?
The integral of (32)/(1-cos(8x)) is -4cot(4x)+C