What is the integral of sin^4(1/2 θ)cos^4(1/2 θ) ?

The integral of sin^4(1/2 θ)cos^4(1/2 θ) is

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integral of sin^4(1/2 θ)cos^4(1/2 θ)

\int sin^{4} (\frac{1}{2} θ) cos^{4} (\frac{1}{2} θ) d θ

2 (- \frac{1}{4} sin^{3} (\frac{θ}{2}) cos (\frac{θ}{2}) + \frac{3}{16} (θ - sin (θ)) - \frac{3}{256} (5 (3 (θ - sin (θ)) - 4 sin^{3} (\frac{θ}{2}) cos (\frac{θ}{2})) - 16 sin^{5} (\frac{θ}{2}) cos (\frac{θ}{2})) - \frac{1}{8} sin^{7} (\frac{θ}{2}) cos (\frac{θ}{2})) + C

Solution steps

\int sin^{4} (\frac{1}{2} θ) cos^{4} (\frac{1}{2} θ) d θ

Simplify

sin^{4} (\frac{1}{2} θ) cos^{4} (\frac{1}{2} θ) : sin^{4} (\frac{θ}{2}) cos^{4} (\frac{θ}{2})

= \int sin^{4} (\frac{θ}{2}) cos^{4} (\frac{θ}{2}) d θ

Apply u-substitution

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

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

Rewrite using trig identities

= 2 \cdot \int sin^{4} (u) (1 - sin^{2} (u))^{2} d u

Expand

sin^{4} (u) (1 - sin^{2} (u))^{2} : sin^{4} (u) - 2 sin^{6} (u) + sin^{8} (u)

= 2 \cdot \int sin^{4} (u) - 2 sin^{6} (u) + sin^{8} (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 sin^{4} (u) d u - \int 2 sin^{6} (u) d u + \int sin^{8} (u) d u)

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

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

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

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

Substitute back

u = \frac{θ}{2}

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

Simplify

2 (- \frac{1}{4} sin^{3} (\frac{θ}{2}) cos (\frac{θ}{2}) + \frac{3}{8} (\frac{θ}{2} - \frac{1}{2} sin (2 \cdot \frac{θ}{2})) - 2 (- \frac{cos ( \frac{θ}{2} ) sin ^{5} ( \frac{θ}{2} )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (\frac{θ}{2}) cos (\frac{θ}{2}) + \frac{3}{8} (\frac{θ}{2} - \frac{1}{2} sin (2 \cdot \frac{θ}{2})))) - \frac{cos ( \frac{θ}{2} ) sin ^{7} ( \frac{θ}{2} )}{8} + \frac{7}{8} (- \frac{cos ( \frac{θ}{2} ) sin ^{5} ( \frac{θ}{2} )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (\frac{θ}{2}) cos (\frac{θ}{2}) + \frac{3}{8} (\frac{θ}{2} - \frac{1}{2} sin (2 \cdot \frac{θ}{2}))))) : 2 (- \frac{1}{4} sin^{3} (\frac{θ}{2}) cos (\frac{θ}{2}) + \frac{3}{16} (θ - sin (θ)) - \frac{3}{256} (5 (3 (θ - sin (θ)) - 4 sin^{3} (\frac{θ}{2}) cos (\frac{θ}{2})) - 16 sin^{5} (\frac{θ}{2}) cos (\frac{θ}{2})) - \frac{1}{8} sin^{7} (\frac{θ}{2}) cos (\frac{θ}{2}))

= 2 (- \frac{1}{4} sin^{3} (\frac{θ}{2}) cos (\frac{θ}{2}) + \frac{3}{16} (θ - sin (θ)) - \frac{3}{256} (5 (3 (θ - sin (θ)) - 4 sin^{3} (\frac{θ}{2}) cos (\frac{θ}{2})) - 16 sin^{5} (\frac{θ}{2}) cos (\frac{θ}{2})) - \frac{1}{8} sin^{7} (\frac{θ}{2}) cos (\frac{θ}{2}))

Add a constant to the solution

= 2 (- \frac{1}{4} sin^{3} (\frac{θ}{2}) cos (\frac{θ}{2}) + \frac{3}{16} (θ - sin (θ)) - \frac{3}{256} (5 (3 (θ - sin (θ)) - 4 sin^{3} (\frac{θ}{2}) cos (\frac{θ}{2})) - 16 sin^{5} (\frac{θ}{2}) cos (\frac{θ}{2})) - \frac{1}{8} sin^{7} (\frac{θ}{2}) cos (\frac{θ}{2})) + C

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\frac{d}{d t} (v (t) e^{0.1 t})

\int x^{8} d x

\int_{0}^{1} x (x + 1) d x

\int \frac{e ^{0}}{( sin ( 3 x ) ) ^{2}} d x

What is the integral of sin^4(1/2 θ)cos^4(1/2 θ) ?
The integral of sin^4(1/2 θ)cos^4(1/2 θ) is