What is the integral of-picsc((pix)/2)cot((pix)/2) ?

The integral of-picsc((pix)/2)cot((pix)/2) is 2csc((pix)/2)+C

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integral of-picsc((pix)/2)cot((pix)/2)

\int - π csc (\frac{π x}{2}) cot (\frac{π x}{2}) d x

2 csc (\frac{π x}{2}) + C

Solution steps

\int - π csc (\frac{π x}{2}) cot (\frac{π x}{2}) d x

Take the constant out:

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

= - π \cdot \int csc (\frac{π x}{2}) cot (\frac{π x}{2}) d x

Apply u-substitution

: \int csc (u) cot (u) \frac{2}{π} d u

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

Take the constant out:

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

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

Apply u-substitution

: \int - 1 d v

= - π \frac{2}{π} \cdot \int - 1 d v

Integral of a constant:

\int a d x = a x

= - π \frac{2}{π} (- 1) v

Substitute back

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

Simplify

- π \frac{2}{π} (- 1) csc (\frac{π x}{2}) : 2 csc (\frac{π x}{2})

= 2 csc (\frac{π x}{2})

Add a constant to the solution

= 2 csc (\frac{π x}{2}) + C

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

\frac{\partial}{\partial x} (8^{x})

\int_{1}^{x} \frac{1}{1 + t ^{2}} d t

e^{x} (3 x^{2} - 5 x)

\frac{d y}{d x} = \frac{( - y ^{2} )}{( 1 + y ^{2} )} \cdot (e^{- x})

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