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

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

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integral of-5cot^3(x)csc^5(x)

\int - 5 cot^{3} (x) csc^{5} (x) d x

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

Solution steps

\int - 5 cot^{3} (x) csc^{5} (x) d x

Take the constant out:

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

= - 5 \cdot \int cot^{3} (x) csc^{5} (x) d x

Apply u-substitution

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

= - 5 \cdot \int - u^{4} (u^{2} - 1) d u

Take the constant out:

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

= - 5 (- \int u^{4} (u^{2} - 1) d u)

Expand

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

= - 5 (- \int u^{6} - u^{4} d u)

Apply the Sum Rule:

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

= - 5 (- (\int u^{6} d u - \int u^{4} d u))

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

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

= - 5 (- (\frac{u ^{7}}{7} - \frac{u ^{5}}{5}))

Substitute back

u = csc (x)

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

Simplify

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

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

Add a constant to the solution

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

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