What is the integral of sin^4(x/2)cos^6(x/2) ?

The integral of sin^4(x/2)cos^6(x/2) is

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integral of sin^4(x/2)cos^6(x/2)

\int sin^{4} (\frac{x}{2}) cos^{6} (\frac{x}{2}) d x

2 (\frac{1}{6} cos^{5} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{5}{96} (4 cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + 3 (x + sin (x))) - \frac{11}{7680} (96 cos^{7} (\frac{x}{2}) sin (\frac{x}{2}) + 7 (16 cos^{5} (\frac{x}{2}) sin (\frac{x}{2}) + 5 (4 cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + 3 (x + sin (x))))) + \frac{1}{10} cos^{9} (\frac{x}{2}) sin (\frac{x}{2})) + C

Solution steps

\int sin^{4} (\frac{x}{2}) cos^{6} (\frac{x}{2}) d x

Apply u-substitution

: \int sin^{4} (u) cos^{6} (u) \cdot 2 d u

= \int sin^{4} (u) cos^{6} (u) \cdot 2 d u

Take the constant out:

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

= 2 \cdot \int sin^{4} (u) cos^{6} (u) d u

Rewrite using trig identities

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

Expand

(1 - cos^{2} (u))^{2} cos^{6} (u) : cos^{6} (u) - 2 cos^{8} (u) + cos^{10} (u)

= 2 \cdot \int cos^{6} (u) - 2 cos^{8} (u) + cos^{10} (u) d u

Apply the Sum Rule:

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

= 2 (\int cos^{6} (u) d u - \int 2 cos^{8} (u) d u + \int cos^{10} (u) d u)

\int cos^{6} (u) d u = \frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u)))

\int 2 cos^{8} (u) d u = 2 (\frac{sin ( u ) cos ^{7} ( u )}{8} + \frac{7}{8} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u)))))

\int cos^{10} (u) d u = \frac{sin ( u ) cos ^{9} ( u )}{10} + \frac{9}{10} (\frac{sin ( u ) cos ^{7} ( u )}{8} + \frac{7}{8} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u)))))

= 2 (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u))) - 2 (\frac{sin ( u ) cos ^{7} ( u )}{8} + \frac{7}{8} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u))))) + \frac{sin ( u ) cos ^{9} ( u )}{10} + \frac{9}{10} (\frac{sin ( u ) cos ^{7} ( u )}{8} + \frac{7}{8} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u))))))

Substitute back

u = \frac{x}{2}

= 2 (\frac{sin ( \frac{x}{2} ) cos ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} + \frac{1}{2} sin (2 \cdot \frac{x}{2}))) - 2 (\frac{sin ( \frac{x}{2} ) cos ^{7} ( \frac{x}{2} )}{8} + \frac{7}{8} (\frac{sin ( \frac{x}{2} ) cos ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} + \frac{1}{2} sin (2 \cdot \frac{x}{2}))))) + \frac{sin ( \frac{x}{2} ) cos ^{9} ( \frac{x}{2} )}{10} + \frac{9}{10} (\frac{sin ( \frac{x}{2} ) cos ^{7} ( \frac{x}{2} )}{8} + \frac{7}{8} (\frac{sin ( \frac{x}{2} ) cos ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} + \frac{1}{2} sin (2 \cdot \frac{x}{2}))))))

Simplify

2 (\frac{sin ( \frac{x}{2} ) cos ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} + \frac{1}{2} sin (2 \cdot \frac{x}{2}))) - 2 (\frac{sin ( \frac{x}{2} ) cos ^{7} ( \frac{x}{2} )}{8} + \frac{7}{8} (\frac{sin ( \frac{x}{2} ) cos ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} + \frac{1}{2} sin (2 \cdot \frac{x}{2}))))) + \frac{sin ( \frac{x}{2} ) cos ^{9} ( \frac{x}{2} )}{10} + \frac{9}{10} (\frac{sin ( \frac{x}{2} ) cos ^{7} ( \frac{x}{2} )}{8} + \frac{7}{8} (\frac{sin ( \frac{x}{2} ) cos ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} + \frac{1}{2} sin (2 \cdot \frac{x}{2})))))) : 2 (\frac{1}{6} cos^{5} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{5}{96} (4 cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + 3 (x + sin (x))) - \frac{11}{7680} (96 cos^{7} (\frac{x}{2}) sin (\frac{x}{2}) + 7 (16 cos^{5} (\frac{x}{2}) sin (\frac{x}{2}) + 5 (4 cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + 3 (x + sin (x))))) + \frac{1}{10} cos^{9} (\frac{x}{2}) sin (\frac{x}{2}))

= 2 (\frac{1}{6} cos^{5} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{5}{96} (4 cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + 3 (x + sin (x))) - \frac{11}{7680} (96 cos^{7} (\frac{x}{2}) sin (\frac{x}{2}) + 7 (16 cos^{5} (\frac{x}{2}) sin (\frac{x}{2}) + 5 (4 cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + 3 (x + sin (x))))) + \frac{1}{10} cos^{9} (\frac{x}{2}) sin (\frac{x}{2}))

Add a constant to the solution

= 2 (\frac{1}{6} cos^{5} (\frac{x}{2}) sin (\frac{x}{2}) + \frac{5}{96} (4 cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + 3 (x + sin (x))) - \frac{11}{7680} (96 cos^{7} (\frac{x}{2}) sin (\frac{x}{2}) + 7 (16 cos^{5} (\frac{x}{2}) sin (\frac{x}{2}) + 5 (4 cos^{3} (\frac{x}{2}) sin (\frac{x}{2}) + 3 (x + sin (x))))) + \frac{1}{10} cos^{9} (\frac{x}{2}) sin (\frac{x}{2})) + C

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What is the integral of sin^4(x/2)cos^6(x/2) ?
The integral of sin^4(x/2)cos^6(x/2) is