What is the integral of (cos^5(2x))/(sqrt(sin(2x))) ?

The integral of (cos^5(2x))/(sqrt(sin(2x))) is sqrt(sin(2x))-2/5 sin^{5/2}(2x)+1/9 sin^{9/2}(2x)+C

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integral of (cos^5(2x))/(sqrt(sin(2x)))

\int \frac{cos ^{5} ( 2 x )}{sin ( 2 x )} d x

sin (2 x) - \frac{2}{5} sin^{\frac{5}{2}} (2 x) + \frac{1}{9} sin^{\frac{9}{2}} (2 x) + C

Solution steps

\int \frac{cos ^{5} ( 2 x )}{sin ( 2 x )} d x

Apply u-substitution

: \int \frac{cos ^{5} ( u )}{2 sin ( u )} d u

= \int \frac{cos ^{5} ( u )}{2 sin ( u )} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{cos ^{5} ( u )}{sin ( u )} d u

Rewrite using trig identities

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

Apply u-substitution

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

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

Expand

\frac{( 1 - v ^{2} ) ^{2}}{v} : \frac{1}{v} - 2 v^{\frac{3}{2}} + v^{\frac{7}{2}}

= \frac{1}{2} \cdot \int \frac{1}{v} - 2 v^{\frac{3}{2}} + v^{\frac{7}{2}} 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

= \frac{1}{2} (\int \frac{1}{v} d v - \int 2 v^{\frac{3}{2}} d v + \int v^{\frac{7}{2}} d v)

\int \frac{1}{v} d v = 2 v

\int 2 v^{\frac{3}{2}} d v = \frac{4}{5} v^{\frac{5}{2}}

\int v^{\frac{7}{2}} d v = \frac{2}{9} v^{\frac{9}{2}}

= \frac{1}{2} (2 v - \frac{4}{5} v^{\frac{5}{2}} + \frac{2}{9} v^{\frac{9}{2}})

Substitute back

= \frac{1}{2} (2 sin (2 x) - \frac{4}{5} sin^{\frac{5}{2}} (2 x) + \frac{2}{9} sin^{\frac{9}{2}} (2 x))

Simplify

\frac{1}{2} (2 sin (2 x) - \frac{4}{5} sin^{\frac{5}{2}} (2 x) + \frac{2}{9} sin^{\frac{9}{2}} (2 x)) : sin (2 x) - \frac{2}{5} sin^{\frac{5}{2}} (2 x) + \frac{1}{9} sin^{\frac{9}{2}} (2 x)

= sin (2 x) - \frac{2}{5} sin^{\frac{5}{2}} (2 x) + \frac{1}{9} sin^{\frac{9}{2}} (2 x)

Add a constant to the solution

= sin (2 x) - \frac{2}{5} sin^{\frac{5}{2}} (2 x) + \frac{1}{9} sin^{\frac{9}{2}} (2 x) + C

sin^{2} (x) cos (x)

\int \frac{x ^{2}}{5 x ^{2} + 12} d x

\int 5 e^{- 0.7 x} d x

\int u u^{2} - 5 d u

\int sin (2 x - 4) d x

What is the integral of (cos^5(2x))/(sqrt(sin(2x))) ?
The integral of (cos^5(2x))/(sqrt(sin(2x))) is sqrt(sin(2x))-2/5 sin^{5/2}(2x)+1/9 sin^{9/2}(2x)+C