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

Lösung

\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

Schritte zur Lösung

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

Vereinfache

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

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

Wende integrale Reduktion an

= - \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}))

Füge eine Konstante zur Lösung hinzu

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

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

\int \frac{x ^{2} + 2}{3} d x

\int \frac{x ^{3} - 1}{( x ^{4} - 4 x ) ^{3}} d x

\int x^{2} sin (y x) d x