What is the integral of 3/(cos(2x)-1) ?

The integral of 3/(cos(2x)-1) is 3/2 cot(x)+C

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integral of 3/(cos(2x)-1)

\int \frac{3}{cos ( 2 x ) - 1} d x

\frac{3}{2} cot (x) + C

Solution steps

\int \frac{3}{cos ( 2 x ) - 1} d x

Take the constant out:

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

= 3 \cdot \int \frac{1}{cos ( 2 x ) - 1} d x

Rewrite using trig identities

= 3 \cdot \int - \frac{1}{2 sin ^{2} ( x )} d x

Take the constant out:

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

= 3 (- \frac{1}{2} \cdot \int \frac{1}{sin ^{2} ( x )} d x)

Use the common integral:

\int \frac{1}{sin ^{2} ( x )} d x = - cot (x)

= 3 (- \frac{1}{2} (- cot (x)))

Simplify

3 (- \frac{1}{2} (- cot (x))) : \frac{3}{2} cot (x)

= \frac{3}{2} cot (x)

Add a constant to the solution

= \frac{3}{2} cot (x) + C

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What is the integral of 3/(cos(2x)-1) ?
The integral of 3/(cos(2x)-1) is 3/2 cot(x)+C