What is the integral of (40)/(1-cos(5x)) ?

The integral of (40)/(1-cos(5x)) is -8cot((5x)/2)+C

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integral of (40)/(1-cos(5x))

\int \frac{40}{1 - cos ( 5 x )} d x

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

Solution steps

\int \frac{40}{1 - cos ( 5 x )} d x

Take the constant out:

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

= 40 \cdot \int \frac{1}{1 - cos ( 5 x )} d x

Apply u-substitution

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

= 40 \cdot \int \frac{1}{5 ( 1 - cos ( u ) )} d u

Take the constant out:

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

= 40 \cdot \frac{1}{5} \cdot \int \frac{1}{1 - cos ( u )} d u

Apply u-substitution

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

= 40 \cdot \frac{1}{5} \cdot \int \frac{1}{v ^{2}} d v

Apply the Power Rule

: - \frac{1}{v}

= 40 \cdot \frac{1}{5} (- \frac{1}{v})

Substitute back

= 40 \cdot \frac{1}{5} (- \frac{1}{tan ( \frac{5 x}{2} )})

Simplify

40 \cdot \frac{1}{5} (- \frac{1}{tan ( \frac{5 x}{2} )}) : - 8 cot (\frac{5 x}{2})

= - 8 cot (\frac{5 x}{2})

Add a constant to the solution

= - 8 cot (\frac{5 x}{2}) + C

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What is the integral of (40)/(1-cos(5x)) ?
The integral of (40)/(1-cos(5x)) is -8cot((5x)/2)+C