What is the integral of csc^{10}(x)cot^3(x) ?

The integral of csc^{10}(x)cot^3(x) is -1/12 csc^{12}(x)+1/10 csc^{10}(x)+C

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integral of csc^{10}(x)cot^3(x)

\int csc^{10} (x) cot^{3} (x) d x

- \frac{1}{12} csc^{12} (x) + \frac{1}{10} csc^{10} (x) + C

Solution steps

\int csc^{10} (x) cot^{3} (x) d x

Rewrite using trig identities

= \int \frac{( 1 - sin ^{2} ( x ) ) cos ( x )}{sin ^{13} ( x )} d x

Apply u-substitution

: \int \frac{1 - u ^{2}}{u ^{13}} d u

= \int \frac{1 - u ^{2}}{u ^{13}} d u

Expand

\frac{1 - u ^{2}}{u ^{13}} : \frac{1}{u ^{13}} - \frac{1}{u ^{11}}

= \int \frac{1}{u ^{13}} - \frac{1}{u ^{11}} 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

= \int \frac{1}{u ^{13}} d u - \int \frac{1}{u ^{11}} d u

\int \frac{1}{u ^{13}} d u = - \frac{1}{12 u ^{12}}

\int \frac{1}{u ^{11}} d u = - \frac{1}{10 u ^{10}}

= - \frac{1}{12 u ^{12}} - (- \frac{1}{10 u ^{10}})

Substitute back

u = sin (x)

= - \frac{1}{12 sin ^{12} ( x )} - (- \frac{1}{10 sin ^{10} ( x )})

Simplify

- \frac{1}{12 sin ^{12} ( x )} - (- \frac{1}{10 sin ^{10} ( x )}) : - \frac{1}{12} csc^{12} (x) + \frac{1}{10} csc^{10} (x)

= - \frac{1}{12} csc^{12} (x) + \frac{1}{10} csc^{10} (x)

Add a constant to the solution

= - \frac{1}{12} csc^{12} (x) + \frac{1}{10} csc^{10} (x) + C

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What is the integral of csc^{10}(x)cot^3(x) ?
The integral of csc^{10}(x)cot^3(x) is -1/12 csc^{12}(x)+1/10 csc^{10}(x)+C