What is the integral of csc^4(5x) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of csc^4(5x)

\int csc^{4} (5 x) d x

\frac{1}{5} (- \frac{1}{3} csc^{3} (5 x) cos (5 x) - \frac{2}{3} cot (5 x)) + C

Solution steps

\int csc^{4} (5 x) d x

Apply u-substitution

: \int csc^{4} (u) \frac{1}{5} d u

= \int csc^{4} (u) \frac{1}{5} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{5} \cdot \int csc^{4} (u) d u

Apply Integral Reduction

= \frac{1}{5} (- \frac{cos ( u ) csc ^{3} ( u )}{3} + \frac{2}{3} \cdot \int csc^{2} (u) d u)

\int csc^{2} (u) d u = - cot (u)

= \frac{1}{5} (- \frac{cos ( u ) csc ^{3} ( u )}{3} + \frac{2}{3} (- cot (u)))

Substitute back

u = 5 x

= \frac{1}{5} (- \frac{cos ( 5 x ) csc ^{3} ( 5 x )}{3} + \frac{2}{3} (- cot (5 x)))

Simplify

\frac{1}{5} (- \frac{cos ( 5 x ) csc ^{3} ( 5 x )}{3} + \frac{2}{3} (- cot (5 x))) : \frac{1}{5} (- \frac{1}{3} csc^{3} (5 x) cos (5 x) - \frac{2}{3} cot (5 x))

= \frac{1}{5} (- \frac{1}{3} csc^{3} (5 x) cos (5 x) - \frac{2}{3} cot (5 x))

Add a constant to the solution

= \frac{1}{5} (- \frac{1}{3} csc^{3} (5 x) cos (5 x) - \frac{2}{3} cot (5 x)) + C

\int 1 - x x^{2} d x

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

\int 7 cos (7 x) d x

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

\int \frac{e ^{5 x}}{2 + e ^{5 x}} d x

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