What is the integral of cotan^4(6t)sec(6t) ?

The integral of cotan^4(6t)sec(6t) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of cotan^4(6t)sec(6t)

\int co tan^{4} (6 t) sec (6 t) d t

\frac{co ( tan ^{2} ( 3 t ) ( tan ( 3 t ) - 1 ) ^{4} + 6 tan ( 3 t ) ( tan ( 3 t ) - 1 ) ^{4} + 4 tan ( 3 t ) ( tan ( 3 t ) + 1 ) ^{4} + 3 ln ∣ tan ( 3 t ) + 1 ∣ ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{4} + 3 ( tan ( 3 t ) - 1 ) ^{4} - 3 ln ∣ tan ( 3 t ) - 1 ∣ ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{4} - 2 ( tan ( 3 t ) + 1 ) ^{4} - 3 ( tan ( 3 t ) + 1 ) ^{3} ( tan ( 3 t ) - 1 ) ^{4} - 3 ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{3} - ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{2} )}{48 ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{4}} + C

Solution steps

\int co tan^{4} (6 t) sec (6 t) d t

Take the constant out:

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

= co \cdot \int sec (6 t) tan^{4} (6 t) d t

Apply u-substitution

: \int sec (u) tan^{4} (u) \frac{1}{6} d u

= co \cdot \int sec (u) tan^{4} (u) \frac{1}{6} d u

Take the constant out:

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

= co \frac{1}{6} \cdot \int sec (u) tan^{4} (u) d u

Rewrite using trig identities

= co \frac{1}{6} \cdot \int \frac{sin ^{4} ( u )}{cos ^{5} ( u )} d u

Apply u-substitution

: \int \frac{32 v ^{4}}{( 1 - v ^{2} ) ^{5}} d v

= co \frac{1}{6} \cdot \int \frac{32 v ^{4}}{( 1 - v ^{2} ) ^{5}} d v

Take the constant out:

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

= co \frac{1}{6} \cdot 32 \cdot \int \frac{v ^{4}}{( 1 - v ^{2} ) ^{5}} d v

Take the partial fraction of

\frac{v ^{4}}{( 1 - v ^{2} ) ^{5}} : \frac{3}{256 ( v + 1 )} + \frac{3}{256 ( v + 1 ) ^{2}} - \frac{1}{128 ( v + 1 ) ^{3}} - \frac{3}{64 ( v + 1 ) ^{4}} + \frac{1}{32 ( v + 1 ) ^{5}} - \frac{3}{256 ( v - 1 )} + \frac{3}{256 ( v - 1 ) ^{2}} + \frac{1}{128 ( v - 1 ) ^{3}} - \frac{3}{64 ( v - 1 ) ^{4}} - \frac{1}{32 ( v - 1 ) ^{5}}

= co \frac{1}{6} \cdot 32 \cdot \int \frac{3}{256 ( v + 1 )} + \frac{3}{256 ( v + 1 ) ^{2}} - \frac{1}{128 ( v + 1 ) ^{3}} - \frac{3}{64 ( v + 1 ) ^{4}} + \frac{1}{32 ( v + 1 ) ^{5}} - \frac{3}{256 ( v - 1 )} + \frac{3}{256 ( v - 1 ) ^{2}} + \frac{1}{128 ( v - 1 ) ^{3}} - \frac{3}{64 ( v - 1 ) ^{4}} - \frac{1}{32 ( v - 1 ) ^{5}} d v

Apply the Sum Rule:

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

= co \frac{1}{6} \cdot 32 (\int \frac{3}{256 ( v + 1 )} d v + \int \frac{3}{256 ( v + 1 ) ^{2}} d v - \int \frac{1}{128 ( v + 1 ) ^{3}} d v - \int \frac{3}{64 ( v + 1 ) ^{4}} d v + \int \frac{1}{32 ( v + 1 ) ^{5}} d v - \int \frac{3}{256 ( v - 1 )} d v + \int \frac{3}{256 ( v - 1 ) ^{2}} d v + \int \frac{1}{128 ( v - 1 ) ^{3}} d v - \int \frac{3}{64 ( v - 1 ) ^{4}} d v - \int \frac{1}{32 ( v - 1 ) ^{5}} d v)

\int \frac{3}{256 ( v + 1 )} d v = \frac{3}{256} ln ∣ v + 1 ∣

\int \frac{3}{256 ( v + 1 ) ^{2}} d v = - \frac{3}{256 ( v + 1 )}

\int \frac{1}{128 ( v + 1 ) ^{3}} d v = - \frac{1}{256 ( v + 1 ) ^{2}}

\int \frac{3}{64 ( v + 1 ) ^{4}} d v = - \frac{1}{64 ( v + 1 ) ^{3}}

\int \frac{1}{32 ( v + 1 ) ^{5}} d v = - \frac{1}{128 ( v + 1 ) ^{4}}

\int \frac{3}{256 ( v - 1 )} d v = \frac{3}{256} ln ∣ v - 1 ∣

\int \frac{3}{256 ( v - 1 ) ^{2}} d v = - \frac{3}{256 ( v - 1 )}

\int \frac{1}{128 ( v - 1 ) ^{3}} d v = - \frac{1}{256 ( v - 1 ) ^{2}}

\int \frac{3}{64 ( v - 1 ) ^{4}} d v = - \frac{1}{64 ( v - 1 ) ^{3}}

\int \frac{1}{32 ( v - 1 ) ^{5}} d v = - \frac{1}{128 ( v - 1 ) ^{4}}

= co \frac{1}{6} \cdot 32 (\frac{3}{256} ln ∣ v + 1 ∣ - \frac{3}{256 ( v + 1 )} - (- \frac{1}{256 ( v + 1 ) ^{2}}) - (- \frac{1}{64 ( v + 1 ) ^{3}}) - \frac{1}{128 ( v + 1 ) ^{4}} - \frac{3}{256} ln ∣ v - 1 ∣ - \frac{3}{256 ( v - 1 )} - \frac{1}{256 ( v - 1 ) ^{2}} - (- \frac{1}{64 ( v - 1 ) ^{3}}) - (- \frac{1}{128 ( v - 1 ) ^{4}}))

Substitute back

= co \frac{1}{6} \cdot 32 (\frac{3}{256} ln tan (\frac{6 t}{2}) + 1 - \frac{3}{256 ( tan ( \frac{6 t}{2} ) + 1 )} - (- \frac{1}{256 ( tan ( \frac{6 t}{2} ) + 1 ) ^{2}}) - (- \frac{1}{64 ( tan ( \frac{6 t}{2} ) + 1 ) ^{3}}) - \frac{1}{128 ( tan ( \frac{6 t}{2} ) + 1 ) ^{4}} - \frac{3}{256} ln tan (\frac{6 t}{2}) - 1 - \frac{3}{256 ( tan ( \frac{6 t}{2} ) - 1 )} - \frac{1}{256 ( tan ( \frac{6 t}{2} ) - 1 ) ^{2}} - (- \frac{1}{64 ( tan ( \frac{6 t}{2} ) - 1 ) ^{3}}) - (- \frac{1}{128 ( tan ( \frac{6 t}{2} ) - 1 ) ^{4}}))

Simplify

co \frac{1}{6} \cdot 32 (\frac{3}{256} ln tan (\frac{6 t}{2}) + 1 - \frac{3}{256 ( tan ( \frac{6 t}{2} ) + 1 )} - (- \frac{1}{256 ( tan ( \frac{6 t}{2} ) + 1 ) ^{2}}) - (- \frac{1}{64 ( tan ( \frac{6 t}{2} ) + 1 ) ^{3}}) - \frac{1}{128 ( tan ( \frac{6 t}{2} ) + 1 ) ^{4}} - \frac{3}{256} ln tan (\frac{6 t}{2}) - 1 - \frac{3}{256 ( tan ( \frac{6 t}{2} ) - 1 )} - \frac{1}{256 ( tan ( \frac{6 t}{2} ) - 1 ) ^{2}} - (- \frac{1}{64 ( tan ( \frac{6 t}{2} ) - 1 ) ^{3}}) - (- \frac{1}{128 ( tan ( \frac{6 t}{2} ) - 1 ) ^{4}})) : \frac{co ( tan ^{2} ( 3 t ) ( tan ( 3 t ) - 1 ) ^{4} + 6 tan ( 3 t ) ( tan ( 3 t ) - 1 ) ^{4} + 4 tan ( 3 t ) ( tan ( 3 t ) + 1 ) ^{4} + 3 ln ∣ tan ( 3 t ) + 1 ∣ ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{4} + 3 ( tan ( 3 t ) - 1 ) ^{4} - 3 ln ∣ tan ( 3 t ) - 1 ∣ ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{4} - 2 ( tan ( 3 t ) + 1 ) ^{4} - 3 ( tan ( 3 t ) + 1 ) ^{3} ( tan ( 3 t ) - 1 ) ^{4} - 3 ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{3} - ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{2} )}{48 ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{4}}

= \frac{co ( tan ^{2} ( 3 t ) ( tan ( 3 t ) - 1 ) ^{4} + 6 tan ( 3 t ) ( tan ( 3 t ) - 1 ) ^{4} + 4 tan ( 3 t ) ( tan ( 3 t ) + 1 ) ^{4} + 3 ln ∣ tan ( 3 t ) + 1 ∣ ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{4} + 3 ( tan ( 3 t ) - 1 ) ^{4} - 3 ln ∣ tan ( 3 t ) - 1 ∣ ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{4} - 2 ( tan ( 3 t ) + 1 ) ^{4} - 3 ( tan ( 3 t ) + 1 ) ^{3} ( tan ( 3 t ) - 1 ) ^{4} - 3 ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{3} - ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{2} )}{48 ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{4}}

Add a constant to the solution

= \frac{co ( tan ^{2} ( 3 t ) ( tan ( 3 t ) - 1 ) ^{4} + 6 tan ( 3 t ) ( tan ( 3 t ) - 1 ) ^{4} + 4 tan ( 3 t ) ( tan ( 3 t ) + 1 ) ^{4} + 3 ln ∣ tan ( 3 t ) + 1 ∣ ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{4} + 3 ( tan ( 3 t ) - 1 ) ^{4} - 3 ln ∣ tan ( 3 t ) - 1 ∣ ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{4} - 2 ( tan ( 3 t ) + 1 ) ^{4} - 3 ( tan ( 3 t ) + 1 ) ^{3} ( tan ( 3 t ) - 1 ) ^{4} - 3 ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{3} - ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{2} )}{48 ( tan ( 3 t ) + 1 ) ^{4} ( tan ( 3 t ) - 1 ) ^{4}} + C

\int cos^{2} (nπ x) d x

\int \frac{x ^{3} + 2 x - 1}{( x - 3 ) ^{3}} d x

\int \frac{1}{( x ^{2} + 1 ) ( x ^{2} + 3 )} d x

\int \frac{4 x ^{3}}{x ^{2} - 9} d x

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

What is the integral of cotan^4(6t)sec(6t) ?
The integral of cotan^4(6t)sec(6t) is