What is the integral of sin^5(2x)cos^{3/2}(2x) ?

The integral of sin^5(2x)cos^{3/2}(2x) is -1/5 cos^{5/2}(2x)+2/9 cos^{9/2}(2x)-1/13 cos^{13/2}(2x)+C

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integral of sin^5(2x)cos^{3/2}(2x)

\int sin^{5} (2 x) cos^{\frac{3}{2}} (2 x) d x

- \frac{1}{5} cos^{\frac{5}{2}} (2 x) + \frac{2}{9} cos^{\frac{9}{2}} (2 x) - \frac{1}{13} cos^{\frac{13}{2}} (2 x) + C

Solution steps

\int sin^{5} (2 x) cos^{\frac{3}{2}} (2 x) d x

Apply u-substitution

: \int sin^{5} (u) cos^{\frac{3}{2}} (u) \frac{1}{2} d u

= \int sin^{5} (u) cos^{\frac{3}{2}} (u) \frac{1}{2} 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 sin^{5} (u) cos^{\frac{3}{2}} (u) d u

Rewrite using trig identities

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

Apply u-substitution

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

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

Expand

- v^{\frac{3}{2}} (1 - v^{2})^{2} : - v^{\frac{3}{2}} + 2 v^{\frac{7}{2}} - v^{\frac{11}{2}}

= \frac{1}{2} \cdot \int - v^{\frac{3}{2}} + 2 v^{\frac{7}{2}} - v^{\frac{11}{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 v^{\frac{3}{2}} d v + \int 2 v^{\frac{7}{2}} d v - \int v^{\frac{11}{2}} d v)

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

\int 2 v^{\frac{7}{2}} d v = \frac{4}{9} v^{\frac{9}{2}}

\int v^{\frac{11}{2}} d v = \frac{2}{13} v^{\frac{13}{2}}

= \frac{1}{2} (- \frac{2}{5} v^{\frac{5}{2}} + \frac{4}{9} v^{\frac{9}{2}} - \frac{2}{13} v^{\frac{13}{2}})

Substitute back

= \frac{1}{2} (- \frac{2}{5} cos^{\frac{5}{2}} (2 x) + \frac{4}{9} cos^{\frac{9}{2}} (2 x) - \frac{2}{13} cos^{\frac{13}{2}} (2 x))

Simplify

\frac{1}{2} (- \frac{2}{5} cos^{\frac{5}{2}} (2 x) + \frac{4}{9} cos^{\frac{9}{2}} (2 x) - \frac{2}{13} cos^{\frac{13}{2}} (2 x)) : - \frac{1}{5} cos^{\frac{5}{2}} (2 x) + \frac{2}{9} cos^{\frac{9}{2}} (2 x) - \frac{1}{13} cos^{\frac{13}{2}} (2 x)

= - \frac{1}{5} cos^{\frac{5}{2}} (2 x) + \frac{2}{9} cos^{\frac{9}{2}} (2 x) - \frac{1}{13} cos^{\frac{13}{2}} (2 x)

Add a constant to the solution

= - \frac{1}{5} cos^{\frac{5}{2}} (2 x) + \frac{2}{9} cos^{\frac{9}{2}} (2 x) - \frac{1}{13} cos^{\frac{13}{2}} (2 x) + C

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What is the integral of sin^5(2x)cos^{3/2}(2x) ?
The integral of sin^5(2x)cos^{3/2}(2x) is -1/5 cos^{5/2}(2x)+2/9 cos^{9/2}(2x)-1/13 cos^{13/2}(2x)+C