What is the integral of 1/(sin^5(x)) ?

The integral of 1/(sin^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 1/(sin^5(x))

\int \frac{1}{sin ^{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 \frac{1}{sin ^{5} ( x )} d x

Use the basic trigonometric identity:

\frac{1}{sin ( x )} = csc (x)

= \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

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