What is the integral of 1024sin^4(x)cos^4(x) ?

The integral of 1024sin^4(x)cos^4(x) is

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integral of 1024sin^4(x)cos^4(x)

\int 1024 sin^{4} (x) cos^{4} (x) d x

1024 (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)) - \frac{9}{8} (- \frac{1}{6} sin^{5} (x) cos (x) + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)))) - \frac{1}{8} sin^{7} (x) cos (x)) + C

Solution steps

\int 1024 sin^{4} (x) cos^{4} (x) d x

Take the constant out:

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

= 1024 \cdot \int sin^{4} (x) cos^{4} (x) d x

Rewrite using trig identities

= 1024 \cdot \int sin^{4} (x) (1 - sin^{2} (x))^{2} d x

Expand

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

= 1024 \cdot \int sin^{4} (x) - 2 sin^{6} (x) + sin^{8} (x) d x

Apply the Sum Rule:

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

= 1024 (\int sin^{4} (x) d x - \int 2 sin^{6} (x) d x + \int sin^{8} (x) d x)

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

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

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

= 1024 (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)) - 2 (- \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)))) - \frac{cos ( x ) sin ^{7} ( x )}{8} + \frac{7}{8} (- \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)))))

Simplify

1024 (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)) - 2 (- \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)))) - \frac{cos ( x ) sin ^{7} ( x )}{8} + \frac{7}{8} (- \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))))) : 1024 (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)) - \frac{9}{8} (- \frac{1}{6} sin^{5} (x) cos (x) + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)))) - \frac{1}{8} sin^{7} (x) cos (x))

= 1024 (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)) - \frac{9}{8} (- \frac{1}{6} sin^{5} (x) cos (x) + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)))) - \frac{1}{8} sin^{7} (x) cos (x))

Add a constant to the solution

= 1024 (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)) - \frac{9}{8} (- \frac{1}{6} sin^{5} (x) cos (x) + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)))) - \frac{1}{8} sin^{7} (x) cos (x)) + C

\int \frac{t ^{5}}{( t ^{3} + 1 ) ^{2}} d t

\frac{d}{d x} (\frac{x ^{n} - 1}{x - 1})

y = x 3600 - x^{2}

2 t^{3} cos (t)

\frac{\partial}{\partial z} (e^{x y z^{2} + z})

What is the integral of 1024sin^4(x)cos^4(x) ?
The integral of 1024sin^4(x)cos^4(x) is