What is the integral of 1/(csc^2(6x)-csc(6x)) ?

The integral of 1/(csc^2(6x)-csc(6x)) is -1/(3(tan(3x)-1))+1/6 cos(6x)-x+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 1/(csc^2(6x)-csc(6x))

\int \frac{1}{csc ^{2} ( 6 x ) - csc ( 6 x )} d x

- \frac{1}{3 ( tan ( 3 x ) - 1 )} + \frac{1}{6} cos (6 x) - x + C

Solution steps

\int \frac{1}{csc ^{2} ( 6 x ) - csc ( 6 x )} d x

Apply u-substitution

: \int \frac{1}{6 ( csc ^{2} ( u ) - csc ( u ) )} d u

= \int \frac{1}{6 ( csc ^{2} ( u ) - csc ( u ) )} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{csc ^{2} ( u ) - csc ( u )} d u

Rewrite using trig identities

= \frac{1}{6} \cdot \int \frac{1 - cos ^{2} ( u )}{1 - sin ( u )} d u

Expand

\frac{1 - cos ^{2} ( u )}{1 - sin ( u )} : \frac{1}{1 - sin ( u )} - \frac{cos ^{2} ( u )}{1 - sin ( u )}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{6} (\int \frac{1}{1 - sin ( u )} d u - \int \frac{cos ^{2} ( u )}{1 - sin ( u )} d u)

\int \frac{1}{1 - sin ( u )} d u = - \frac{2}{tan ( \frac{u}{2} ) - 1}

\int \frac{cos ^{2} ( u )}{1 - sin ( u )} d u = - cos (u) + u

= \frac{1}{6} (- \frac{2}{tan ( \frac{u}{2} ) - 1} - (- cos (u) + u))

Substitute back

u = 6 x

= \frac{1}{6} (- \frac{2}{tan ( \frac{6 x}{2} ) - 1} - (- cos (6 x) + 6 x))

Simplify

\frac{1}{6} (- \frac{2}{tan ( \frac{6 x}{2} ) - 1} - (- cos (6 x) + 6 x)) : - \frac{1}{3 ( tan ( 3 x ) - 1 )} + \frac{1}{6} cos (6 x) - x

= - \frac{1}{3 ( tan ( 3 x ) - 1 )} + \frac{1}{6} cos (6 x) - x

Add a constant to the solution

= - \frac{1}{3 ( tan ( 3 x ) - 1 )} + \frac{1}{6} cos (6 x) - x + C

\int (x^{3} - 3 x^{2} + 4) e^{2 - 3 x} d x

\int \frac{1}{9} e^{- 3 x} d x

\int \frac{1}{8 w ^{9}} d w

\int y^{3} 3 - y^{2} d y

\int \frac{3 x - 9}{( x + 4 ) ( x - 5 )} d x

What is the integral of 1/(csc^2(6x)-csc(6x)) ?
The integral of 1/(csc^2(6x)-csc(6x)) is -1/(3(tan(3x)-1))+1/6 cos(6x)-x+C