What is the integral of csc^5(pix)cot^3(pix) ?

The integral of csc^5(pix)cot^3(pix) is -1/pi (1/7 csc^7(pix)-1/5 csc^5(pix))+C

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integral of csc^5(pix)cot^3(pix)

\int csc^{5} (π x) cot^{3} (π x) d x

- \frac{1}{π} (\frac{1}{7} csc^{7} (π x) - \frac{1}{5} csc^{5} (π x)) + C

Solution steps

\int csc^{5} (π x) cot^{3} (π x) d x

Apply u-substitution

: \int csc^{5} (u) cot^{3} (u) \frac{1}{π} d u

= \int csc^{5} (u) cot^{3} (u) \frac{1}{π} d u

Take the constant out:

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

= \frac{1}{π} \cdot \int csc^{5} (u) cot^{3} (u) d u

Apply u-substitution

: \int - v^{4} (v^{2} - 1) d v

= \frac{1}{π} \cdot \int - v^{4} (v^{2} - 1) d v

Take the constant out:

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

= \frac{1}{π} (- \int v^{4} (v^{2} - 1) d v)

Expand

v^{4} (v^{2} - 1) : v^{6} - v^{4}

= \frac{1}{π} (- \int v^{6} - v^{4} d v)

Apply the Sum Rule:

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

= \frac{1}{π} (- (\int v^{6} d v - \int v^{4} d v))

\int v^{6} d v = \frac{v ^{7}}{7}

\int v^{4} d v = \frac{v ^{5}}{5}

= \frac{1}{π} (- (\frac{v ^{7}}{7} - \frac{v ^{5}}{5}))

Substitute back

= \frac{1}{π} (- (\frac{csc ^{7} ( π x )}{7} - \frac{csc ^{5} ( π x )}{5}))

Simplify

\frac{1}{π} (- (\frac{csc ^{7} ( π x )}{7} - \frac{csc ^{5} ( π x )}{5})) : - \frac{1}{π} (\frac{1}{7} csc^{7} (π x) - \frac{1}{5} csc^{5} (π x))

= - \frac{1}{π} (\frac{1}{7} csc^{7} (π x) - \frac{1}{5} csc^{5} (π x))

Add a constant to the solution

= - \frac{1}{π} (\frac{1}{7} csc^{7} (π x) - \frac{1}{5} csc^{5} (π x)) + C

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What is the integral of csc^5(pix)cot^3(pix) ?
The integral of csc^5(pix)cot^3(pix) is -1/pi (1/7 csc^7(pix)-1/5 csc^5(pix))+C