What is the integral of sin^6(x) ?

The integral of sin^6(x) is -(cos(x)sin^5(x))/6+5/6 (-1/4 sin^3(x)cos(x)+3/8 (x-1/2 sin(2x)))+C

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integral of sin^6(x)

\int sin^{6} (x) d x

- \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))) + C

Solution steps

\int sin^{6} (x) d x

Apply Integral Reduction

= - \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} \cdot \int sin^{4} (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))

= - \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)))

Add a constant to the solution

= - \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))) + C

x \to 0 lim (\frac{1}{4 x ^{2}})

f (x) = 1.8 sin (x) - x

\int \frac{15}{( x - 1 ) ( x ^{2} + 4 )} d x

f (x) = e^{5 x}

\frac{d y}{d x} = \frac{40}{x ^{- 5}}

What is the integral of sin^6(x) ?
The integral of sin^6(x) is -(cos(x)sin^5(x))/6+5/6 (-1/4 sin^3(x)cos(x)+3/8 (x-1/2 sin(2x)))+C