What is the integral of (10)/(sec(2x)-1) ?

The integral of (10)/(sec(2x)-1) is 5(-2x-cot(x))+C

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integral of (10)/(sec(2x)-1)

\int \frac{10}{sec ( 2 x ) - 1} d x

5 (- 2 x - cot (x)) + C

Solution steps

\int \frac{10}{sec ( 2 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}{sec ( 2 x ) - 1} d x

Apply u-substitution

: \int \frac{1}{2 ( sec ( u ) - 1 )} d u

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

Take the constant out:

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

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

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{sec ( u )}{sec ( u ) - 1} d u = - cot (\frac{u}{2})

\int 1 d u = u

= 10 \cdot \frac{1}{2} (- cot (\frac{u}{2}) - u)

Substitute back

u = 2 x

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

Simplify

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

= 5 (- 2 x - cot (x))

Add a constant to the solution

= 5 (- 2 x - cot (x)) + C

f (x) = \frac{ln ( x )}{x - 1}

\int (2 t + 7)^{2.9} d t

3 x^{3} - 5 x + 7 x^{4} - 9 - x^{2}

f (x) = - 3 x^{2} + 4 x + 20

x \to 3 lim (\frac{x - 5}{x + 3})

What is the integral of (10)/(sec(2x)-1) ?
The integral of (10)/(sec(2x)-1) is 5(-2x-cot(x))+C