What is the integral of (-4)/(cos(x)-1) ?

The integral of (-4)/(cos(x)-1) is -4cot(x/2)+C

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integral of (-4)/(cos(x)-1)

\int \frac{- 4}{cos ( x ) - 1} d x

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

Solution steps

\int \frac{- 4}{cos ( x ) - 1} d x

Take the constant out:

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

= - 4 \cdot \int \frac{1}{cos ( x ) - 1} d x

Apply u-substitution

: \int - \frac{1}{u ^{2}} d u

= - 4 \cdot \int - \frac{1}{u ^{2}} d u

Take the constant out:

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

= - 4 (- \int \frac{1}{u ^{2}} d u)

Apply the Power Rule

: - \frac{1}{u}

= - 4 (- (- \frac{1}{u}))

Substitute back

u = tan (\frac{x}{2})

= - 4 (- (- \frac{1}{tan ( \frac{x}{2} )}))

Simplify

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

= - 4 cot (\frac{x}{2})

Add a constant to the solution

= - 4 cot (\frac{x}{2}) + C

6 (ln (y)) y^{^{'}} - x^{3} y = 0

x \to \frac{π}{2} lim (9 sec (x))

x \to \infty lim (\frac{1}{x} - 2)

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

\int (x - 2)^{2} d x

What is the integral of (-4)/(cos(x)-1) ?
The integral of (-4)/(cos(x)-1) is -4cot(x/2)+C