What is the integral of cot^8(x/3)csc^2(x/3) ?

The integral of cot^8(x/3)csc^2(x/3) is -1/3 cot^9(x/3)+C

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integral of cot^8(x/3)csc^2(x/3)

\int cot^{8} (\frac{x}{3}) csc^{2} (\frac{x}{3}) d x

- \frac{1}{3} cot^{9} (\frac{x}{3}) + C

Solution steps

\int cot^{8} (\frac{x}{3}) csc^{2} (\frac{x}{3}) d x

Apply u-substitution

: \int cot^{8} (u) csc^{2} (u) \cdot 3 d u

= \int cot^{8} (u) csc^{2} (u) \cdot 3 d u

Take the constant out:

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

= 3 \cdot \int cot^{8} (u) csc^{2} (u) d u

Apply u-substitution

: \int - v^{8} d v

= 3 \cdot \int - v^{8} d v

Take the constant out:

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

= 3 (- \int v^{8} d v)

Apply the Power Rule

: \frac{v ^{9}}{9}

= 3 (- \frac{v ^{9}}{9})

Substitute back

= 3 (- \frac{cot ^{9} ( \frac{x}{3} )}{9})

Simplify

3 (- \frac{cot ^{9} ( \frac{x}{3} )}{9}) : - \frac{1}{3} cot^{9} (\frac{x}{3})

= - \frac{1}{3} cot^{9} (\frac{x}{3})

Add a constant to the solution

= - \frac{1}{3} cot^{9} (\frac{x}{3}) + C

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What is the integral of cot^8(x/3)csc^2(x/3) ?
The integral of cot^8(x/3)csc^2(x/3) is -1/3 cot^9(x/3)+C