What is the integral of (cos^{23}(x))/(sin(x)) ?

The integral of (cos^{23}(x))/(sin(x)) is

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integral of (cos^{23}(x))/(sin(x))

integral

\frac{cos ^{23} ( x )}{sin ( x )}

ln ∣ sin (x) ∣ - \frac{11 sin ^{2} ( x )}{2} + \frac{55 sin ^{4} ( x )}{4} - \frac{55 sin ^{6} ( x )}{2} + \frac{165 sin ^{8} ( x )}{4} - \frac{231 sin ^{10} ( x )}{5} + \frac{77 sin ^{12} ( x )}{2} - \frac{165 sin ^{14} ( x )}{7} + \frac{165 sin ^{16} ( x )}{16} - \frac{55 sin ^{18} ( x )}{18} + \frac{11 sin ^{20} ( x )}{20} - \frac{sin ^{22} ( x )}{22} + C

Solution steps

\int \frac{cos ^{23} ( x )}{sin ( x )} d x

Rewrite using trig identities

= \int \frac{( 1 - sin ^{2} ( x ) ) ^{11} cos ( x )}{sin ( x )} d x

Apply u-substitution

= \int \frac{( 1 - u ^{2} ) ^{11}}{u} d u

Expand

\frac{( 1 - u ^{2} ) ^{11}}{u} : \frac{1}{u} - 11 u + 55 u^{3} - 165 u^{5} + 330 u^{7} - 462 u^{9} + 462 u^{11} - 330 u^{13} + 165 u^{15} - 55 u^{17} + 11 u^{19} - u^{21}

= \int \frac{1}{u} - 11 u + 55 u^{3} - 165 u^{5} + 330 u^{7} - 462 u^{9} + 462 u^{11} - 330 u^{13} + 165 u^{15} - 55 u^{17} + 11 u^{19} - u^{21} 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} d u - \int 11 u d u + \int 55 u^{3} d u - \int 165 u^{5} d u + \int 330 u^{7} d u - \int 462 u^{9} d u + \int 462 u^{11} d u - \int 330 u^{13} d u + \int 165 u^{15} d u - \int 55 u^{17} d u + \int 11 u^{19} d u - \int u^{21} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int 11 u d u = \frac{11 u ^{2}}{2}

\int 55 u^{3} d u = \frac{55 u ^{4}}{4}

\int 165 u^{5} d u = \frac{55 u ^{6}}{2}

\int 330 u^{7} d u = \frac{165 u ^{8}}{4}

\int 462 u^{9} d u = \frac{231 u ^{10}}{5}

\int 462 u^{11} d u = \frac{77 u ^{12}}{2}

\int 330 u^{13} d u = \frac{165 u ^{14}}{7}

\int 165 u^{15} d u = \frac{165 u ^{16}}{16}

\int 55 u^{17} d u = \frac{55 u ^{18}}{18}

\int 11 u^{19} d u = \frac{11 u ^{20}}{20}

\int u^{21} d u = \frac{u ^{22}}{22}

= ln ∣ u ∣ - \frac{11 u ^{2}}{2} + \frac{55 u ^{4}}{4} - \frac{55 u ^{6}}{2} + \frac{165 u ^{8}}{4} - \frac{231 u ^{10}}{5} + \frac{77 u ^{12}}{2} - \frac{165 u ^{14}}{7} + \frac{165 u ^{16}}{16} - \frac{55 u ^{18}}{18} + \frac{11 u ^{20}}{20} - \frac{u ^{22}}{22}

Substitute back

u = sin (x)

= ln ∣ sin (x) ∣ - \frac{11 sin ^{2} ( x )}{2} + \frac{55 sin ^{4} ( x )}{4} - \frac{55 sin ^{6} ( x )}{2} + \frac{165 sin ^{8} ( x )}{4} - \frac{231 sin ^{10} ( x )}{5} + \frac{77 sin ^{12} ( x )}{2} - \frac{165 sin ^{14} ( x )}{7} + \frac{165 sin ^{16} ( x )}{16} - \frac{55 sin ^{18} ( x )}{18} + \frac{11 sin ^{20} ( x )}{20} - \frac{sin ^{22} ( x )}{22}

Add a constant to the solution

= ln ∣ sin (x) ∣ - \frac{11 sin ^{2} ( x )}{2} + \frac{55 sin ^{4} ( x )}{4} - \frac{55 sin ^{6} ( x )}{2} + \frac{165 sin ^{8} ( x )}{4} - \frac{231 sin ^{10} ( x )}{5} + \frac{77 sin ^{12} ( x )}{2} - \frac{165 sin ^{14} ( x )}{7} + \frac{165 sin ^{16} ( x )}{16} - \frac{55 sin ^{18} ( x )}{18} + \frac{11 sin ^{20} ( x )}{20} - \frac{sin ^{22} ( x )}{22} + C

What is the integral of (cos^{23}(x))/(sin(x)) ?
The integral of (cos^{23}(x))/(sin(x)) is