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

The integral of csc^5(x) is -(cos(x)csc^4(x))/4+3/4 (-1/2 cot(x)csc(x)+1/2 ln|tan(x/2)|)+C

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integral of csc^5(x)

\int csc^{5} (x) d x

- \frac{cos ( x ) csc ^{4} ( x )}{4} + \frac{3}{4} (- \frac{1}{2} cot (x) csc (x) + \frac{1}{2} ln tan (\frac{x}{2})) + C

Solution steps

\int csc^{5} (x) d x

Apply Integral Reduction

= - \frac{cos ( x ) csc ^{4} ( x )}{4} + \frac{3}{4} \cdot \int csc^{3} (x) d x

\int csc^{3} (x) d x = - \frac{1}{2} cot (x) csc (x) + \frac{1}{2} ln tan (\frac{x}{2})

= - \frac{cos ( x ) csc ^{4} ( x )}{4} + \frac{3}{4} (- \frac{1}{2} cot (x) csc (x) + \frac{1}{2} ln tan (\frac{x}{2}))

Add a constant to the solution

= - \frac{cos ( x ) csc ^{4} ( x )}{4} + \frac{3}{4} (- \frac{1}{2} cot (x) csc (x) + \frac{1}{2} ln tan (\frac{x}{2})) + C

\frac{4 x}{( x ^{3} - 3 ) ^{2}}

x + tan (x)

\int \frac{x ^{2}}{x ^{2} + y ^{2}} d y

\int x \frac{1}{1} d x

\frac{d}{d t} (b sin (wt))

What is the integral of csc^5(x) ?
The integral of csc^5(x) is -(cos(x)csc^4(x))/4+3/4 (-1/2 cot(x)csc(x)+1/2 ln|tan(x/2)|)+C