What is the integral of (10)/(cos(x)-1) ?

The integral of (10)/(cos(x)-1) is 10cot(x/2)+C

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integral of (10)/(cos(x)-1)

\int \frac{10}{cos ( x ) - 1} d x

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

Solution steps

\int \frac{10}{cos ( x ) - 1} d x

Take the constant out:

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

= 10 \cdot \int \frac{1}{cos ( x ) - 1} d x

Apply u-substitution

= 10 \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

= 10 (- \int \frac{1}{u ^{2}} d u)

Apply the Power Rule

= 10 (- (- \frac{1}{u}))

Substitute back

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

= 10 (- (- \frac{1}{tan ( \frac{x}{2} )}))

Simplify

10 (- (- \frac{1}{tan ( \frac{x}{2} )})) : 10 cot (\frac{x}{2})

= 10 cot (\frac{x}{2})

Add a constant to the solution

= 10 cot (\frac{x}{2}) + C

What is the integral of (10)/(cos(x)-1) ?
The integral of (10)/(cos(x)-1) is 10cot(x/2)+C