What is the integral of csc(1-2x)cot(1-2x) ?

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

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integral of csc(1-2x)cot(1-2x)

\int csc (1 - 2 x) cot (1 - 2 x) d x

\frac{1}{2} csc (1 - 2 x) + C

Solution steps

\int csc (1 - 2 x) cot (1 - 2 x) d x

Apply u-substitution

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

Take the constant out:

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

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

Apply u-substitution

= - \frac{1}{2} \cdot \int - 1 d v

Integral of a constant:

\int a d x = a x

= - \frac{1}{2} (- 1) v

Substitute back

= - \frac{1}{2} (- 1) csc (1 - 2 x)

Simplify

= \frac{1}{2} csc (1 - 2 x)

Add a constant to the solution

= \frac{1}{2} csc (1 - 2 x) + C

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