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

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

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integral of (sin^4(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)) + C

Solution steps

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

Apply Integral Reduction

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

\int sin^{2} (x) d x = \frac{1}{2} (x - \frac{1}{2} sin (2 x))

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

Simplify

- \frac{cos ( x ) sin ^{3} ( x )}{4} + \frac{3}{4} \cdot \frac{1}{2} (x - \frac{1}{2} sin (2 x)) : - \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))

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

x \to 1 lim (2 x^{2} - 4 x + 7)

\frac{10}{16 s ^{2} + 5 s + 1}

\frac{\partial}{\partial x} (x^{2} + 4 y^{2} + 5 z^{2})

f (x) = \frac{7}{10} x^{2} + 4 x^{3}

\frac{d p}{p - p ^{2}} = d t

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