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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integral Reduction

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

Substitute back

u = 4 x

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

Simplify

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

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

Add a constant to the solution

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

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

\int \frac{100}{25 + ( 5 y ) ^{2}} + e^{4} d y

\frac{y ( 4 x + 4 y )}{( x ^{2} + y ^{2} )}

\int_{1}^{4} x^{2} + 2 d x

\frac{\partial}{\partial x} (2 x^{3 y})

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