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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= 5 \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

= 5 (- \int \frac{1}{u ^{2}} d u)

Apply the Power Rule

: - \frac{1}{u}

= 5 (- (- \frac{1}{u}))

Substitute back

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

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

Simplify

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

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

Add a constant to the solution

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

\int y cos (\frac{x}{y}) d x

f (x) = 2 (sin^{3} (x))

\int \frac{sin ( 2 x )}{14 + cos ^{2} ( x )} d x

y^{^{'}} + \frac{2}{x} y = y^{2}, y (1) = 3

\frac{d}{d x} (\frac{a}{2 y})

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