What is the integral of (csc^2(x))^2csc(x) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (csc^2(x))^2csc(x)

\int (csc^{2} (x))^{2} csc (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^{2} (x))^{2} csc (x) d x

Simplify

(csc^{2} (x))^{2} csc (x) : csc^{5} (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

\int csch (2 z) d z

\int \frac{1}{v ^{2} + 5 v - 6} d v

\int e^{\frac{1}{5} l n (x)} d x

\int \frac{12}{( x - 4 ) ( x + 8 )} d x

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

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