What is the integral of-picsc((pix)/8)cot((pix)/8) ?

The integral of-picsc((pix)/8)cot((pix)/8) is 8csc((pix)/8)+C

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integral of-picsc((pix)/8)cot((pix)/8)

\int - π csc (\frac{π x}{8}) cot (\frac{π x}{8}) d x

8 csc (\frac{π x}{8}) + C

Solution steps

\int - π csc (\frac{π x}{8}) cot (\frac{π x}{8}) 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}{8}) cot (\frac{π x}{8}) d x

Apply u-substitution

: \int csc (u) cot (u) \frac{8}{π} d u

= - π \cdot \int csc (u) cot (u) \frac{8}{π} d u

Take the constant out:

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

= - π \frac{8}{π} \cdot \int csc (u) cot (u) d u

Apply u-substitution

: \int - 1 d v

= - π \frac{8}{π} \cdot \int - 1 d v

Integral of a constant:

\int a d x = a x

= - π \frac{8}{π} (- 1) v

Substitute back

= - π \frac{8}{π} (- 1) csc (\frac{π x}{8})

Simplify

- π \frac{8}{π} (- 1) csc (\frac{π x}{8}) : 8 csc (\frac{π x}{8})

= 8 csc (\frac{π x}{8})

Add a constant to the solution

= 8 csc (\frac{π x}{8}) + C

\int (\frac{1}{1 + x ^{2}}) d x

\int \frac{1}{x ^{2} - 10} d x

\int e^{- 0.3 x} d x

cos (x^{2}) sin (x)

\frac{x - 5}{9 x ^{2} e ^{x}}

What is the integral of-picsc((pix)/8)cot((pix)/8) ?
The integral of-picsc((pix)/8)cot((pix)/8) is 8csc((pix)/8)+C